The Quantum Thomist

Musings about quantum physics, classical philosophy, and the connection between the two.
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The empiricist philosophy against Quantum Physics
Last modified on Sun Jun 17 09:14:23 2018


Introduction

I recently finished a series of posts outlining the basis of quantum field theory. These posts offer a basic and simplified (albeit not simple) outline of the scientific basis of my work. It is important to understand them, because everything else that I do builds on that. In the last of those posts, I suggested that Quantum physics was broadly consistent with Aristotelian metaphysics. Not completely consistent, of course: we have to make some modifications to Aristotle's world-view, most importantly his reluctance to view physics mathematically (although, to be fair to him, the mathematics needed to represent causality and space and time is well beyond what was available in his time). I don't claim that Aristotle's philosophy is the end of the discussion, only that it offers a better beginning for the search for a philosophy consistent with contemporary physics than the numerous alternatives that arose following the intellectual revolution that gave birth to Galilean and Newtonian physics.

However, it is not just enough to show consistency. One must also consider rival possibilities, and show them inconsistent with modern physics.

Here, however, I am at a disadvantage. Two disadvantages, in fact. Firstly, I am not a philosopher. Physics is my area of expertise, and what I what little I do know tends be in the area of classical philosophy. Secondly to eliminate all alternatives means going over a lot of ideas, an inhumanly impossible task. So take what I say here as somewhat provisional, and I would welcome correction and enhancements; anything I have got wrong or missed.

I began with the mechanical philosophy, or the philosophy of classical science from Galileo's time until Einstein. It might be thought from my critical remarks that I am wholly opposed to mechanism; but that does it a disservice. I believe the mechanists took a number of wrong turns; but they did at least ask the right questions, and correctly sought a deeper understanding of nature beyond mere sense impression. And there were some things (such as that it is possible and useful to represent space and time geometrically) which the mechanists got right and the classical philosophers wrong. The mechanical philosophy ultimately arose from a medieval Aristotelian world-view. It took a different path, denying the importance of causality, teleology, and natural powers while enforcing ideas such as determinism and the indestructibility of fundamental corpuscles. Well, it got those things wrong. But, nonetheless, it still thought in terms of natural laws which were both rational, intelligible, and which, given enough effort, we could come to understand and know. In that respect, it still shared much of a common language with the older philosophy. The mechanists and Aristotelians can be thought of as distant cousins. They are interested in the same questions, and use similar methodology of coming to the truth through reason and careful observation, but, due to their differing assumptions, came to different answers. But other modern philosophies stand diametrically opposed to them both.

It is all Descartes' fault, of course. He divided the cosmos into the mental world and the physical world, with a chasm between them. Mechanists concentrate on the physical world, and say that our minds ultimately arise from physics. Aristotelians deny the chasm, and most will say that our minds are closely linked to physics, albeit with a distinctly immaterial aspect to them (this union between the material and immaterial is straight-forward in Aristotle's world-view: everything is a union between an immaterial form and a material matter). Both of these philosophies thus say that we should concentrate on the natural sciences first of all, and let our general philosophy arise from that. If there is a mental world, ultimately we would come to understand it through our understanding of the physical world. Physics thus comes first, and mind (at least in our case) is derived in full or in part from physics. We thus can't understand our minds unless we first understand physics, and the other natural sciences built upon physics.

But most other modern philosophies started by thinking about the mental world. They took the opposite approach. They stated that we first need to understand the minds and how we gain understanding. That we can't understand physics unless we first understand our minds. Indeed, many argue that physics is ultimately a product of our minds (or maybe an illusion of our minds) rather than a genuine aspect of reality (or a genuine representation of an aspect of reality). By a product of our minds, I don't mean the notation and so on we use to describe physics (which is a certain extent arbitrary), but to deny that the model we have corresponds to something in reality; it is merely a system we have invented to explain various sensations which also exist in our minds but which is not a representation of various rules that the material world has to follow. Moreover, in this view not only is our current understanding of physics inadequate, it is impossible in principle to map the conclusions of our physical models back onto reality.

One reason why this approach was taken was that mechanism copes badly with the philosophy of the mind. It is very difficult to account for things such as free will, consciousness, and determinate thinking in a mechanistic world-view. Yet it would take a fool, or someone who let his ideology blind him to his experience, to deny such things. Yet, at that time, it seemed as though the physical world was governed by mechanical principles. The way around this that people chose was to say that our mind was non-physical. This led directly to Cartesian dualism, and once Descartes introduced the idea of a separation between the mental and the physical, Descartes' successors, even those of radically different traditions such as the empiricists, still held onto that separation. Descartes' theory of knowledge began with an investigation of the mind (Cogito ergo Sum), and subsequent philosophers also took the investigation of the mental world as primary, even if they didn't accept that the mental world is entirely separate from the physical.

I believe the approach of beginning by trying to understand ourselves to be wholly flawed, since we are part of the physical world, and our brains work by various electro-chemical processes which give rise to our thoughts, senses and memory, and thus we need physics and the chemical and biological sciences to explain all this. Only after we understand the easier scientific part should we think about the mental aspects of our understanding; otherwise, like all philosophical systems detached from experimental knowledge, the chances of us going wrong are too great. But that hasn't stopped many people from taking the path of trying to understanding our minds before trying to understand how our minds work.

That doesn't mean that our model of physics is perfect. It isn't. But, like everything else, the key is not to think in terms of accuracy alone, but both accuracy and precision. Some aspect of natural science we understand very well; we are both accurate and precise. In other aspects we are unsure, but we acknowledge our uncertainty; maybe we have to choose between different possible models; maybe we can only say "I don't know what the true theory is, but it has to satisfy these conditions and reduce to these theories in these limits." In both cases, our limitation is not one of inaccuracy but one of imprecision. The argument that because our theories do not, and cannot, correspond in every tiny detail with perfect precision to the real world and so therefore cannot be representation of reality fails because it erroneously demands that our knowledge must be perfectly precise. They demand that we must make an inference to the best explanation, when the correct attitude is to refuse to make any choices between explanations until sufficient evidence is available. To be imprecisely right is better than being precisely wrong. The philosophies which start from mental processes generally have an incorrect understanding of uncertainty; and thus they fail to properly distinguish between imprecision and inaccuracy. Indeed, from what I have read, they often fail to consider acknowledged imprecision in our mental constructions at all. Which makes them the exact opposite of being scientific in their approach.

Anyway, today I want to talk about empiricism. More specifically, one particular formulation of empiricism, namely that of Locke. Much of what I write will also carry forward to other empiricists, and other systems of thought influenced by empiricism, but I only have space in this post to address this first of the major empiricist philosophers.

Empiricism is the philosophy that our knowledge of the universe comes entirely or mostly from observation. Strictly speaking it is a theory of epistemology (what we can know) rather than a theory of metaphysics (what things truly are), however it does also have implications that affect metaphysics.

Empiricism is one of three basic extreme epistemological positions; the other two are rationalism (we can only know through our reason) and scepticism (we can't know anything). Rationalism in particular states that we have certain innate ideas, from which ultimately all our knowledge arises, by building on them (for example) through syllogisms and logic. Empiricism can be thought of as the denial of such ideas.

Of course, one can also (as I do) take a middle ground; it is possible to be empirical (seeing observation as an important source of our knowledge) without being an empiricist (it is the only important source of our knowledge). For example, Aquinas himself argued that all our knowledge begins with the senses; but he was no empiricist because he also believed that one could abstract from this knowledge to begin a rational tradition. Rather than starting from specific details of each experiment and working his way down the inverted pyramid of knowledge (which was the empiricist approach), he would take general observations from the senses, such as the existence of change or that man is a rational animal. Then he would ask himself what change or rationality involve precisely, and work upwards from that. Francis Bacon in particular criticised this approach, saying that it had and would make no progress. There is (according to Bacon) too big a leap from observation to the general principles.

My own methodology is that there is room for both methods. While all knowledge has to be compared against reality through experiment and observation (otherwise our philosophy is just a fantasy romance, and usually not a very good one at that), there are some general principles we can draw which do have profound consequences. For some things, there is certainly too big a leap from observation to full understanding. For example, one cannot stare at an ingot of gold and immediately come to profound conclusions concerning its cystal and atomic structure. One has to work gradually, slowly developing and understanding the tools to go to the next level of discussion. But, on the other hand, there are examples where general observations can lead us to general principles which are important to initially frame our philosophy. For example, that things change; that there are three space and one time dimension; that reality is not deterministic, and so on. However, there is no reason why we cannot also simultaneously work from the top through detailed experiment. We go down from the top through experiment and up from the bottom through reason built on general principles; if we do it well the two approaches will meet somewhere in the middle.

What I would rule out, however, is Plato's approach; the purely rationalist. This believes that through reason alone we can come to an understanding of the underlying ideals of the forms; in part because the soul pre-existed before it become embodied and thus retained some innate knowledge of the true world of the forms. No. I do not believe that we can start from our own existence and reason our way to the existence of rice pudding and income tax; there are simply too many logically consistent possible universes, and without observation to start us off and guide us we can't tell which one we are in.

Second Part of the Introduction

Before I begin with Locke, it is worth introducing a little basic set theory, which can be a useful analogy to identify the most important difference between the empiricist approach and my own approach.

There are two main objects of study in set theory: sets and mappings. A set is a collection of (usually related) objects. For example, all possible real numbers taken together form a set. All fruits form another set. Our internal representation of fruits form another set. A mapping is a relationship between sets. That is to say, a mapping takes each member of one set and relates it to a member of another set. For example, a mathematical function is a mapping. The function f(x) = x2 is a mapping between a set containing all real numbers (called the domain) and another set containing all real numbers (called the co-domain). The range is the set of members of the co-domain which are identified with a member of the domain via the mapping. In this case, the range is the set of all positive real numbers, so the co-domain is different from the range.

There are different types of mapping. The first is known as surjective. This is where every member of the co-domain is mapped to at least one member of the domain. An injective mapping is where every member of the domain is mapped to a different member of the co-domain. It is bijective if it is both injective and surjective, i.e. every member of the domain is mapped to a single and unique member of the co-domain. These functions are known as invertible, because it is also possible to get from every member of the co-domain to a unique member of the co-domain.

For example (if the domain and co-domain are both the set of all real numbers), the function f(x) = x2 is neither injective or surjective. It is not injective because two different members of the domain (for example 2 and -2) map to the same member of the co-domain (in this case 4). It is not surjective because there are elements of the co-domain (for example -2) which are not reached in the mapping. Of course, this conclusion depends on the definition of the domain and co-domain as much as it does on the function.

On the other hand, the function f(x) = x3 (if the domain containing the members denoted by x and the co-domain containing the members denoted by f(x) are both the set of all real numbers) is both surjective and injective: every real number (domain) is mapped to another real number (co-domain); every real number in the co-domain is mapped to once and only once.

Epistemology is the study of the relationship between two sets; firstly the set of all physical objects, and secondly the set of our own internal image or representation of those objects. Now it is clear that our image of something is not the same as the thing itself. My image of an olive tree is something in my brain, and, since I don't have any trees growing inside me (or I would be very worried if I did) it is clear that my image is not the same as the tree itself. But nonetheless, there is some relationship between the thing and my image of it. After all, we gather information about the tree through our senses, and our image of the tree is influenced by that sensual data.

Epistemology is concerned with the mapping between the physical world and our representation of it. While most philosophers who studied epistemology concentrated on the mechanism of that mapping, it is worth taking a step back first of all and note that there are in principle four different classes of epistemologies. Firstly, we have the two way mapping, where we can both map from reality to the representation and from the representation to reality, without any loss of information. Secondly, there is the one way mapping, where we can move from the reality to the representation, but not the other way round. Then there is the one way mapping where we can move from the representation to reality. And finally there is the zero way mapping, where reality and representation are unrelated to each other. Assuming that we keep the reality and our mental representation of it distinct from each other, these are the only four options.

It is also clear that the tree itself is not a simple thing. After all, the tree is constantly growing; even if the rate of growth is not perceptible to us. Therefore the single being of the tree cannot be thought of as a set containing one single member; it is a set containing each possible state the tree can be in; from the first sprouting seed to the full grown tree. Likewise, if we are wise, our image of the tree would not be a single static thing, but a set of images corresponding to the different stages of growth of the tree. And this is true not only for organic matter, but inorganic matter. Even in a lump of rock, electrons are continually moving from one energy band to another. The atoms are always vibrating. It is in constant motion, and therefore cannot be represented by a single state. Again, if we are wise, our mental image of the rock would not be an unique static idea, but a set of distinct states.

So therefore, we have a mapping between the set containing all possible states of a physical object and the set containing all possible states in our mental representation of that object. And the question I want to pose is whether that mapping is injective, surjective, both or neither. (Of course, there are also other issues concerning accuracy and the precision of our mapping; I leave these aside here so I can concentrate on the most important point, namely the nature of the mapping itself).

From what I can see, most philosophers, particularly since the enlightenment and especially in the empiricist and idealist traditions have stated that the mapping is not injective; in other words that the co-domain of our mental representations contains fewer members than present in the set containing the physical object. These philosophies primarily focus on observable properties of the object rather than its states; so they consider the set of all properties of the object rather than (as I have been doing) the set of all states. They basically say that, while we gather up much information through our senses, we cannot gather up all the information contained in the tree. Our mental representation of the tree is therefore necessarily deficient and incomplete. This means that the mapping is only one way; we can draw from nature to get a mental image, but we cannot go back from our mental image to compare against the physical object again. Therefore, in some senses, understanding our ideas of reality cannot give us an understanding of reality itself. Reality is a closed book to us.

However, these philosophers go even further than that. They don't only claim that their own way of representing reality is not injective. That is something proved relatively easily, and using their methods I believe that it is so. But they also implicitly claim that no possible representation of reality is injective. In other words, we are incapable of mapping back from our mental representation of reality to reality itself. This statement, to my mind, is both far more difficult to prove and far more troubling.

If true, it would mean that we could never come to a knowledge of reality. We would never be able to make predictions; or at least could never be confident of our predictions, since each prediction we make would be built up from our own images, and we can't get from them to reality. It would mean that the study of physics is a dead end. Whatever model we make of the universe cannot hope to be accurate; we could never come to a complete understanding of reality, nor make predictions.

The success of physics (and the rest of the natural sciences), however, suggests that the empiricists are incorrect. It is possible to have a bijective and thus invertible representation of at least some aspects of reality; we can make accurate predictions through considering how our abstract representations of the universe change in time. The reason why the empiricist philosophers failed is that they used the wrong methods to make the mapping from reality to representation. They didn't consider mathematical representations; they didn't consider representations of states rather than representations built from properties. Indeed, I note that, unlike the early mechanists (who were usually great mathematicians), the early empiricists were uniformly rather poor mathematicians and ignorant of the ways of mathematical thinking.

And this, of course, has important implications for metaphysics. When we abstract to the representation of something, we add things to it. We express it in a certain framework or logical structure. That structure is not part of the thing in itself; just something we have used to understand it. The notion of potentia in Aristotle's philosophy are one such structure; the quantum states of modern physics another. There are, of course, numerous different formalisms we could use; if accurate, then each of them would be equivalent in that they are convertible (we never say that something is at location x. We say that it is at location x in some given coordinate system. It might be at location y in a different coordinate system. But that is no contradiction; the two statements are instead entirely equivalent because we know how to convert statements expressed in one coordinate system to another and that x in the first system represents the same physical point as y in the other). We then manipulate our representation so we can understand it, and then project the result of such manipulation back to the real world. That projection cannot contain any reference to the formalism we have created. That restricts the possible types of formalism we use. But since everything expressed in our formalism is in a one to one relationship with something in reality, that restricts what reality could be. In physics, this restriction manifests itself in various symmetries, which (since symmetry is a feature of the real world as our abstractions) must also be manifest in reality.

Thus we have two main differences between the scientific and Aristotelian epistemologies on one hand and the empirical epistemology on the other. The scientific and Aristotelian approaches construct sets representing states of being while the empiricists concentrate on sets of properties or predicates. The scientific and Aristotelian approaches believe that it is possible to construct a two-way mapping relating our representation of reality with reality itself, the empiricist believes that the mapping is only one way (from reality to representation, via our senses and induction). That is not to say that the scientist or Aristotelian philosopher would claim to possess with complete accuracy or precision the correct bijective model of reality; merely that it is in principle possible to find one, and they are going to try to find it, and have made much progress already.

I have drawn a couple of pictures which may or may not illustrate the differences between the two approaches. First of all, a representation of Locke's beliefs about human understanding. Obviously, there is far more that could be said than this; I have just focussed on those strands which are important to my discussion. Probably my most important omission is the importance of memory. And you will have to excuse my complete incompetence when it comes to anything artistic.

A little explanation is needed. The line on the left represents reality; the line on the right represents our mental images, or ideas as Locke calls them. The first step is to obtain sensual data from our sight, touch, and so on. These land in our heads as numerous disparate simple ideas. We combine together these simple ideas to form complex ideas; among these are ideas of number, of space, time, substance and causes. By combining these together, we get a model of physics. Note that these arrows are all one-way. Each arrow represents a mental process which either subtracts information or which combines that information with additional superpositions. For example, causality is not (in this model) something we observe, but something we invent to explain the common conjunction of various more simple ideas. Substance again is a notion we invent to explain the common conjunctions of various basic observed properties or predicates. All we observe is the conjunction. Anything beyond this does not come from our sensual data, and is thus (according to the empiricists) inadmissible. We thus combine, simplify and add to our simple ideas. Similarly, our notion of substance does not correspond to anything in the real world. It is therefore impossible to get back from our understanding to reality. Therefore, all the arrows are one way. Theories of physics describe our ideas, but not reality.

Now we come to my own philosophy, and this chart is a bit more complicated. Again, this is somewhat over-simplified, and I have again left a lot out. My most important omission is that none of the arrows should be thought of as representing a perfectly precise process. There is imprecision in each step, which only increases as we move further down the chain. However that imprecision can be systematically parametrised using Bayesian probability.

Unlike Locke, I have two sources of knowledge. The first is reality; the second is various logical principles; the principle of self contradiction; the assumption that the universe is intelligible, and so on. I have added to this the metaphysical assumptions. Everybody has metaphysical assumptions, and in particular they influence how we interpret the basic data from our senses. Those who claim to have a system with no metaphysical import are merely claiming without evidence that their own assumptions are self-evident, and they might not have even given them any thought. Invariably the metaphysics of those who claim to not need metaphysics is incoherent and poorly thought out, and the source of many other errors. In the middle is our mental representation of reality. I prefer the world representation rather than idea to convey that what is manifested in our thoughts has (or can have) a degree of equivalence to what is in reality. In group theory, representations are different but equivalent ways of expressing the same data or structure; I use the word in the same sense. Each representation follows the same rules of manipulation; for every way of manipulating the members of one representation, there is an equivalent in the other. Our symbol 2 is not the same thing as the number, but it nonetheless represents that number and we subject our representation to the same basic rules that govern the number (for example, how it changes under addition and subtraction); the representative symbol and reality are thus distinct but equivalent.

Once again, we start from reality, and move from reality to our mind via sensation. We receive this as a coherent image or noise or touch, not broken down into disparate parts. We interpret this image using our metaphysical assumptions, and this gives us our most basic level of representation. I have fed in our metaphysics and logic here, but in practice they affect every mental process we have.

From this representation, we can go in two ways. The first is in a non-scientific (that is, non-mathematical) direction. This isn't necessarily wrong (indeed, in some fields of study it is most definitely the right way to go), it just isn't of interest to my discussion here.

The second approach is the scientific one. Here we are interested in creating a mathematical representation of what we see. Only mathematics is precise enough to do what we need, and therefore we make use of it. The first step is to map our experience of reality to a geometrical structure. We need additional information to do this: first of all, our knowledge of geometry; secondly, from reality, basic information needed to constrain the geometry, such as the number of space and time dimensions, the topological and metrical structure of the universe, and so on. This is a different type of information than our basic sensual perception of individual objects, so it gets its own line. The line is two way because in the process of creating the geometrical representation, we ask questions of reality, and then get information back in the form of answers. We are not interested in generalities here, but something very specific required by the representation. The answers are "yes/no"; unlike when we see a cloud and get an image of it in our minds, there is no (obvious) room for ambiguity.

The next step is to get an algebraic representation; the means to move from a geometrical to algebraic representation of space and time were introduced by Descartes, but we can now extend those methods to also cover actual and potential states of being, and thus Aristotelian form. These are the quantum states of the system. At this point, we again need to get some information from experiment to fill in some of the gaps (for example how many types of particle there are, their masses and charges and so on); again, this is a two way process because we are looking for specific answers and theory guides what and how we search for those answers. Finding patterns between different types of particle helps determine underlying symmetries of the theory. This step involves a lot of input from various fields of pure mathematics, and for us to make some arbitrary choices such as a choice of coordinate system. An algebraic representation of reality has more degrees of freedom than reality itself; some of its data is redundant.

Once we have a algebraic framework in place to represent reality, we can then start thinking about the dynamics of the theory. Some experimental input is again useful here, but as I have outlined in previous posts, a lot of this can be calculated from the needs of self-consistency and respecting symmetry requirements. We now have our theory of fundamental science, and from this we can head off to the other sciences.

Finally, we use our physical theory to go back and fine tune the original metaphysical assumptions. If a drastic change to the metaphysics is needed, then we might need to repeat the whole process, until we get something which works.

But the point is that most of these arrows are two-way. The process is reversible. We get from reality to our first representation through sensation, but we can also compare our more developed understanding against reality. We can map back from our algebraic representation to the geometrical representation and to the basic representation. From there we compare against reality; for example by making predictions about things we did not observe and perhaps could not have observed. If those predictions then turn out to be right, then great. If wrong, then we have more data to start our next cycle of refinement of the theory. How is this possible when we introduce additional degrees of freedom? Because when we do so, we are careful to ensure that those aspects of the representation will have no effect on the results when we map back to the previous level. Also note that unlike Locke, where adding a more complex system of ideas required removing information or adding additional ideas haphazardly from our imagination, here we add information carefully so that whatever we put in to help us understand has no effect on the final results. There is uncertainty in the final theory, but it is parametrised, its limits understood, and through the process of refinement we can continually reduce that uncertainty.

The Empiricist Philosophers

The main empiricist philosophers (of the Western European tradition; I neglect ancient Greek and Indian predecessors) were Locke, Berkeley, and Hume. There were fore-runners such as Francis Bacon, and various Italian writers of the Renaissance and medieval Islamic writers. But the empiricist tradition properly formed is usually thought of as beginning with Locke. He is the one who laid down its basic assumptions, although it took Hume to draw those assumptions to their natural conclusions.

Now, there is an important point to consider before looking at the philosophers, and that is to consider the science of their time. Aristotle's physics was now known to have failed. Why? Because it disagreed with experiment. For example, it gave an incorrect account of projectile and pendulum motion in earthly physics, and supernovae and the precise movements of the planets in heavenly physics. By the late middle ages, it became clear that Aristotle's physics, claimed by its detractors to be bottom-up (i.e. based on deduction from underlying philosophical principles) needed replacing with something else.

The immediate successor to Aristotle's physics was the medieval mathematical tradition. This tradition made progress in understanding pendulums, projectiles and so on, but it wasn't at the time moulded into a fully consistent system. It largely died out with the black death, and I think subsequently only de Soto in Salamanca made important work in this area until the time of Galileo. Galileo, of course, was the prime example of the middle ground of epistemology: conducting careful experiments, while searching for the mechanical laws that underlay what he was seeing. Thus it was possible to see Galileo's success as coming primarily from his observation and only secondarily from his mathematical philosophy. The mathematical approach didn't start to yield dramatic successes until Newton's time. But Newton was a contemporary of Hume, and thus lived after the basic foundations of the empiricist philosophy had been laid.

This left the second strand of late medieval and Renaissance science (although, unlike the mathematical physicists, these scholars had fore-runners in the Arab literature, albeit not as careful or precise as the Europeans), the experimental tradition. Rather than dynamics and kinematics, this focussed largely on topics such as magnetism and optics. The natural philosophers would conduct detailed experiments under subtly different conditions, carefully record their findings, and attempt to come up with "laws" explaining the correlations. And these studies had some success. For example, the discovery of the lens, and with it the spectacles, microscopes, and telescopes in the late medieval and Renaissance period. Also the invention of new types of machines, such as the weight driven clock, agricultural revolutions, and improvements to gunpowder based weaponry.

We should thus think about the scientific laws of the time. Let's start with Kepler's laws of motion for the solar system. Kepler had access to Tycho Brahe's detailed tabulated observations of the solar system. He believed that Corpernicus was broadly correct in his principles, but knew that he was wrong in the details. So he tried fitting various mathematical functions to the data, until he found some that worked. Some of his ideas we now know to be direct consequences of Newton's law of gravity, others were just coincidental. But his work was groundbreaking, and derived wholly from observation.

Next, let's consider Snell's law of refraction; linking the angle of incidence of light into a glass block with the angle of transmission via a simple trigonometric relationship. This was derived in different ways by different people, but William Snellius first got it by measuring a load of angles and trying mathematical functions until he found something that worked. Of particular interest to me is Fermat's deviation from the principle of least time. This is, in fact, the penultimate step on the route taken to derive the law from first principles; but in Fermat's day (and more to the point the early empiricist's day) there was no obvious reason why this principle should hold. Even the principle of least time was accepted just from empirical principle: one fit the data to various possibilities, and find something that matches.

Hookes' law that tension of a spring is proportional to extension. Again, first derived by making careful measurements, plotting them on a graph, and noting that they lay in a straight line. Boyle's law of gasses, that the pressure times the volume of a gas is constant at constant temperature (or even its extension to the ideal gas equation, PV=nRT pressure times volume is the number of particles times the temperature times a constant of proportionality; no matter which gas one applies it to -- within a certain approximation, that the gas molecules don't interact with each other except through collisions, that holds well for most gases in sensible conditions). Again experimentally derived, and at that time only experimentally derived.

And that's just physics, and before considering the advances in biology and the early forays into chemistry. Aristotle had tried to argue from the mixed approach, and failed. The mathematicians had argued from a rational approach, and had seemingly made little progress. The empirical scientists had uncovered numerous scientific laws. And these laws, while expressed in terms of basic mathematical relationships, were taking measurements and fitting them to a mathematical function. Their efforts resulted in numerous technological breakthroughs. There was no bottom up reasoning here. It was all top down.

It is important to remember that when the empiricists speak of a law of physics, such as Hume in his argument against miracles, it is this sort of law to which they refer, because they knew no other. We live in a post-Newtonian world, and thus have adopted a post-Newtonian mindset and terminology. But the early empiricists didn't have that advantage. We tend to think in terms of Newton's laws, or Einstein's, or perhaps the Schroedinger equation or laws of statistical mechanics. These are very different beasts. The empiricist laws of physics are specialised to a particular experimental set-up. The mechanical laws are intended to be general. They start from a basic principle, such as the conservation of energy (one of the assumptions behind statistical mechanics), and deduce the consequences of that principle, and only at the end do they test the principle by comparing the consequences against experiment. They do not come directly from experiment. The pure empiricist thus cannot regard them as any more than an interesting curiosity. This is a problem for us, because when we think of laws of physics, our minds turn immediately to the mechanical laws (or perhaps the quantum laws) rather than the directly empirical ones. We read the words of the empiricist writers, and place our own definition into them. This can make it easy for us to misunderstand them.

I cannot address the whole of the empiricist tradition in one blog post, which is already too long, so I will focus on one writer, Locke, and in particular book II of his Essay Concerning Human Understanding. I choose Locke because as the first of the great empiricist philosophers, he set the direction for what followed later, and this section of his writing, because here he outlines the basic premises of his work, and also discusses matters which are significant for theoretical physics, such as space, time, substance and accident. Locke also has the advantage (especially for me) that he is a skilled writer, and reasonably clear in the structure and direction of his thoughts. [I am not looking forward to having to tackle Immanuel Kant later on.]

Ideas and sensation

Locke's presentation

Locke begins with his definition of an idea, as the object of our thoughts. He lists things such as whiteness, hardness, sweetness, thinking, motion, man, elephants and so on as ideas. Note that for Locke such ideas are all in the mind. Whiteness and hardness are not so much properties of external objects, but our reactions to certain sensations. Our conception of the elephant is a purely mental construct. There is a physical being out there which is the source of our sensations, but all we can know is the mental image constructed from that sensation. We can think about that image, take it apart, analyse the parts, put it back together again, do all the things that scientists do, but it will only help us to understand the idea, not the elephant in itself. We can't go back from what we have learnt about the idea, and apply it to the actual elephant. Beyond the raw sense data, the elephant remains a mystery to us, however much we understand the idea of the elephant that it generated.

Locke's main question is how ideas originate. He disagrees with the concept of innate ideas, including the principle of non-contradiction ("It is impossible for the same thing to both be and not be"), on the grounds that this is not universally known; not so much that there are people who disagree with it, but because there are people who are not aware of it.

Locke continues by saying that all ideas come from sensation or reflection. The mind comes to be as a white paper, void of all characters. The ideas in it then cannot come from itself, but only from outside, namely from experience. The means of obtaining ideas from external sources is sensation. Note once again, that ideas are in Locke's thought solely internal to ourselves. From our sensations we derive an impression, and that impression leads to an idea. So our ideas of yellow, white, heat, cold, soft, hard, bitter, sweet and so on go direct from our senses into our mind.

Secondly, once our minds have this primary information, it can begin reflecting on it, and from these reflections derive more complex ideas by combining the basic ideas of sensation together. Locke claims that everything we believe comes from either sensation or reflection.

The mind thinks in proportion to the matter it gets from experience to think about. Follow a child from its birth, and observe the alterations that time makes, and you shall find, as the mind by the senses comes more and more to be furnished with ideas, it comes to be more and more awake; thinks more, the more it has matter to think on. After some time it begins to know the objects which, being most familiar with it, have made lasting impressions. Thus it comes by degrees to know the persons it daily converses with, and distinguishes them from strangers; which are instances and effects of its coming to retain and distinguish the ideas the senses convey to it. And so we may observe how the mind, by degrees, improves in these; and advances to the exercise of those other faculties of enlarging, compounding, and abstracting its ideas, and of reasoning about them, and reflecting upon all these; of which I shall have occasion to speak more hereafter.

A man begins to have ideas when he first has sensation. If it shall be demanded then, when a man begins to have any ideas, I think the true answer is — when he first has any sensation. For, since there appear not to be any ideas in the mind before the senses have conveyed any in, I conceive that ideas in the understanding are coeval with sensation; which is such an impression or motion made in some part of the body, as produces some perception in the understanding. It is about these impressions made on our senses by outward objects that the mind seems first to employ itself, in such operations as we call perception, remembering, consideration, reasoning, etc.

All our knowledge ultimately arises from these reflections. Firstly we reflect on the ideas of sensation; then on these, and gradually building up a tower of knowledge.

Comments

I find Locke's claim about innate ideas such as the principle of non contradiction rather dubious. Certainly people might not have heard of that particularly way of expressing the concept, but the concept itself is evident to everyone who has come to understand the ideas of being and not being. Indeed just understanding the concept of not is enough to draw the conclusion.

This doesn't mean that the principle is innate in the same sense that Locke means it to be. What I want to suggest is that the principle is (albeit we should express it better by adding the clarifications simultaneously, locally, and in the same sense -- something can both be hot and not be cold, or be hot in one place and not hot in another). However, I would describe the principle as self-evident, in that everyone who hears it and understands the meaning of each term would be insane to disagree with it. It is also self-evident in the sense that the statement follows immediately from the definitions of the terms.

So I would not fully object to Locke's contention that we have no innate ideas, but neither do I fully agree with it. He is, I think, asking the wrong question. He asks "Where do our ideas come from?" And in this sense, the principle of self-contradiction is not innate, since it depends on the concepts of being and non-being. But alternatively, we could ask "What is the basis for us to believe certain ideas to be true?" Here we have Locke's same categories of sensation and reflection, but we can also allow some principles to be innate, in that their truth justifies itself. And this, of course, is one of the main criticisms of Locke's method; he does not make the distinction between true ideas and false ideas, or admit to degrees of uncertainty in his epistemology.

To my mind, Locke's beginning here combines some elements of truth with some oversimplifications and omissions. Firstly, he implies that our sensations are from sources wholly external to ourselves. What about self-awareness (which we may term participatory knowledge)? To put it bluntly, I think, therefore I am. For the act of thinking itself is similar to a sensation, regardless of what we are thinking about; but it is not a sensation, since it does not arise from our senses, and nor is it a reflection in Locke's sense of the word because it does not arise solely from the impulses from our senses. Is this principle alone enough to make us aware of everything about the world? No. Is it enough to make us aware of some things? Not least the concept of being. Yes. We understand what it means to be a rational animal, indeed that at least some rational animals exist, because we ourselves are rational animals. This is not information we gain from our senses or from our reflections, but ideas that arise just from being who we are. We need no external stimulus to find such ideas (although observation allows us to expand the concepts of rationality and animality beyond ourselves to others who share those properties). So to say that we can have no ideas (as Locke defines the term) without sensation is surely false. Our ideas would be limited, but not absent.

Secondly, Locke makes no mention of the importance of the precision and imprecision of our ideas, particularly our ideas concerning substances. This is of particular importance when we think that, for example, the term elephant covers a wide variety of individual states of being. As the diligent reader will know, that the same type of being can exist in many different states, is central to modern physics. By failing to recognise this, Locke makes an important omission. Instead, Locke seems to treat the ideas as though they can contain no internal variation.

Thirdly, Locke makes no mention of how our internal ideas are related to those objects which ultimately cause our sensations. For the Aristotlean, the form we grasp in our intellect is, if accurate, the same as the form that is manifested in external objects. I personally take this in a slightly different direction with my thoughts on representations that are in a bijective mapping to the original form. This is surely a matter of the utmost importance. Is the idea of "yellow" solely in our mind, or does it ultimately reside in the banana itself, with our impression a reflection of the banana's yellowness? We know that the colour of an object arises from the structure of its energy bands. The energies of the emitted (and thus reflected) photons depend on the minimum gaps between the excited and ground states of the electrons in the compounds in the surface of the banana; and our distinction of colour is (to simplify a bit) a measurement of the energies of the photons hitting our retinas. Thus our sensation is a representation of something that exists in the banana. Thus we rightly say that "The banana is yellow" (meaning that the energy bands have a particular structure) rather than "our impression of the banana is yellow" (meaning that the idea of yellowness resides only in our mind). Notice also how the notion of energy bands presupposes that bananas have different states or potentia. The banana with the electron in an exited state is in one state; it then emits the photon and resides in the ground state.

Fourthly, even in the reception of simple ideas, there is a certain amount of reflection and abstraction involved. The mind processes even pure sense data.

Fifthly, Locke and the empiricist tradition in general move qualities from reality to our own senses. Colour, for example, ceases to be an aspect of the object, or the light (as the physicist will describe it), but only a matter of our processing of our sensual data. The problem is not that Locke considers the mental aspect of the issue, but that he entirely neglects the physical part of it. To say (using an example from Bertrand Russell's Problem's of Philosophy, which adopts the Lockean position) that light is something we perceive ahead of its nature as a wave motion misses the point: the physical account of light is primary, and our perception a secondary response to that. If they are to be distinguished, they should be referred to by different names. Plus, of course, our brains themselves are physical objects subject to the same sort of laws as photons. We should be explaining our sensual impressions in terms of the underlying biology, and, through that, the underlying physics. Locke's approach gets everything backwards. (While I hold that there are aspects of the mind which are immaterial, I do not mean by that the same thing as Descartes and Locke did. Every physical object has an immaterial and abstractable aspect -- its form -- and our minds are just a particularly interesting example of that. But physics is, as much as anything, the study of particular forms; so I see no reason that the mind should not fall under the purview of physics. But again, I don't mean that in the same sense that the pure materialists do, because they deny any teleology or directionality in physics.)

Sixthly, Locke has no conception of the importance of specifically mathematical abstraction. That is, the ability to represent something by mathematical or geometrical structures; for example space-time by a Euclidean or hyperbolic geometrical space. This is important, because the very act of representation introduces rules on what representations are possible; we don't want, when we map back from the representation to compare against reality, artefacts of our abstraction to interfere with our conclusions about reality. Locke has no knowledge of coordinate systems, and how different complementary systems can represent the same data; equivalent in the sense that they can be converted from one to the other. This is again important; it is fundamental to Einstein's relativity and also other areas of modern physics, and I used it in field theory when showing that all matter must be either Fermions or Bosons. This thinking about different coordinate systems and how they relate to each other is not from a direct reflection on our sensual impressions; nor is it a matter of pure geometry since it relates to our models which are meant to have physical significance (we only have good reason to suppose that our abstractions can be useful if they are a representation of something that is genuinely present in reality).

And this is the most important limitation of Locke's method of epistemology: he leaves out logical and mathematical reflection, or at best treats it as a separate category which can't be linked to the real world. True, the starting points for arithmetic are integers, and such things as addition and subtraction, which we abstract from the real world. But what then? Real numbers, negative numbers, complex numbers, and so on, I suppose you could argue arise from reflections on the integers and basic arithmetic operations. But then what about algebra, functional theory, set theory, calculus, group theory, differential geometry and so on? These don't arise directly from the concept of number, but are abstractions from it. To get from number and addition to algebra, we need another premise, namely generalised abstraction. The process of abstraction adds another set of premises, which don't arise from sense data, which restrict what abstractions are possible. Then, of course, we have the applicability of such logical and mathematical methods to the real world. The two foundational assumptions, the self-consistency of nature, and that it is possible for us to construct bijective representations of it mentally and thus understand it, themselves restrict what nature could be like, and our understanding of it.

So then we have a jump in Locke's thinking. He starts of with the definition of an idea as the object of our thoughts. But then he goes on to describe the simple and primary ideas as what we get from the initial processing of our raw sensual data. He changes the definition midway through his argument; restricting the object of our thoughts to just this one type of thinking. His theory of the development of understanding from children to adulthood is, of course, over generalised: different people get similar conceptions in different ways. For example, I have never been to Abu Dabai, and my conception of the Burj Khalifa is thus only taken through the authorities of others; while those who have seen it will have a different means of understanding it; and those who studied the original architectural blueprints and designed the building another. Yet we are all thinking of the same thing. In the same way, some people come to an understanding of number in the way Locke describes; others through the abstract language and symbols of mathematics in their schooling, or on the authority of the teachers. I have the conception of one thousand and seventeen objects not because I have seen one thousand and seventeen objects (and been aware of it), nor because I have seen one object and extrapolated, but through the abstract process of addition. His belief that we first have ideas (as in objects of our thinking, rather than objects of our sensation) when we first have sensation depends on this shift in definition. The best Locke can do is to argue that the two definitions are equivalent; and this, of course, he attempted to do. But such attempts must fail; there is far more that we can imagine or access through abstract reasoning or representation than can be based on our senses. I can't see a quantum particle such as an electron; I can't touch it or taste it. I can't even say that an electron is like the things I can see, touch or taste (such as shoes, or bullets) only smaller, because it isn't. What I see are classical beings, and a quantum being is fundamentally different. Yet I can understand quantum physics. I come to this understanding via a process which rests on both sensation and abstract reasoning based on innate principles. But this type of idea simply does not fit into Locke's framework. Therefore Locke's framework is deficient.

So Locke's theory of knowledge is good in as far as it goes, but makes several omissions. Sensation and reflection upon sensual impressions are not our only sources of information. That might be true for many animals, but we also have available to us logical and mathematical reasoning, and in particular to apply such reasoning to our representations of external objects. We observe that change occurs in the world, and yet there seems to be continuation through the change. We are left with three options; 1) to say that change is an illusion; 2) to say that continuation is an illusion (and that, for example, I was a different person when I wrote the previous paragraph to when I wrote this one); 3) to accept Aristotle's theory of act and potency (or some variation of it, for example to incorporate quantum mechanical superposition). Once we accept Aristotle's theory, we are led to a concept the set of all potentia relating to a particular individual (which is at least part of what Aristotle meant by form), and the principle of matter which represents that aspect of nature which can't be so abstracted (since our image of a conifer tree is not the same thing as a conifer tree; although it must accurately represent some part of the tree if we are careful and wise enough). By further considering the nature of change, we are led to the notion of efficient and final causality, and so on. Change, of course, is not something we infer directly from our senses; though it might be obvious after we reflect on sensual data. Every one of our thoughts is a movement from one potentia to another; we observe change in ourselves before we even start to sense things. The innate concept of change (though most of us first appreciate it subconsciously) is a pre-requisite to to the concept of sensation, let alone the sensation of any one particular thing. But to get from the existence of change to Aristotle's metaphysics requires a process of logical reasoning which is certainly valid but goes beyond Locke's categories of how we obtain our ideas. Thus, though in practice we come to understanding after sensation, and our sensual data guides us, in principle certain concepts can arise without sensation, and do not depend on sensation, but come before sensation in the strictest ordering of human understanding.

Thus Locke, and the empiricists in general, are wrong that all our "ideas" are built from sensual data and reflections on that data. Logical and mathematical reasoning, the formation of hypotheses, the application of these hypotheses to the physical world, and the testing and elimination of these hypotheses against observation are all important sources of knowledge that the empiricists don't consider. There is a difference between testing against observation and induction from observation. Both are empirical, but the empiricist only believes the latter is reasonable, while the more capable person will deploy them both in the appropriate circumstances. The need for logical and mathematical consistency gives us additional premises not derived from observation which we can combine with our observational data to reach more profound conclusions.

Much of theoretical physics today is tested against observation, but arises from thoughts about the process of mathematical abstraction of the universe. If Locke was right, then such methods would be invalid, and contemporary theoretical physics groundless. The success of physics proves Locke's empiricism (and that of Locke's successors) wrong.

Of modes, substances and relations

Locke's presentation

Simple ideas are those which we passively absorb from our senses. However, from these we develop more complex ideas through reflection. Complex ideas are of one of three types: modes, substances, and relations.

Modes are seen as dependent on, or affections of, substances. Locke gives the examples of triangles, gratitude and murder. These are abstract constructions. Triangles are a particular shape, i.e. a property shared by many objects, but combined from the simple properties we detect directly such as linearity or colour. Gratitude a feeling or reaction we have. Murder is a particular type of act. So I take it that by modes, Locke means things such as properties, emotions or aspects of character and acts. Modes might be simple, such as number, or complex, such as the concept of theft, which relies on various previous concepts such as that of private property, or beauty, which is related to how various aspects of something are put together.

Substances are combinations of simple ideas to represent distinct particular things subsisting in themselves. Note that in Locke's view substances are made from combinations of properties.

Thus if to substance be joined the simple idea of a certain dull whitish colour, with certain degrees of weight, hardness, ductility, and fusibility, we have the idea of lead; and a combination of the ideas of a certain sort of figure, with the powers of motion, thought and reasoning, joined to substance, the ordinary idea of a man.

Substances can refer to singular objects or collections of them.

Thirdly, we have ideas of relation, which arise from comparing one idea to another.

The simplest complex ideas arise from joining together two simple ideas; but we can build on these to combine more obtuse objects.

My comments

There is much to question in this section of Locke's writing (such as whether he is right to lump all the distinct types of mode together, and whether these are indeed the only three possible types of complex ideas; for example the Aristotelian form and quantum mechanical eigenstate or operator don't fit into any of these categories, which suggests that the categories are not broad enough to allow for abstractions and representations).

But I would emphasise in particular Locke's main error here, which has been repeated by numerous philosophers who followed him (and I am thinking in particular of Kant). He reduces our idea of substance to combinations of properties. In other words, he reduces our idea of an elephant to its colour, the shape of its ears, its size, weight and tusks, the way it moves and so on. But the issue is which comes first, the properties or the substance? Is a substance no more than a combination of properties (the way of induction), or are properties something we can deduce from knowledge of the substance (the scientific method of hypothesis testing).

The answer from quantum physics is clear: substance comes first. Take, for example, the metal of lead. Ultimately the theory best used to describe lead is an effective field theory. We start with some basic experimental data (not strictly necessary; but saves us time by directing us to the right substance): the atomic number and weight of lead, and the crystalline structure of the arrangement of its nuclei. From these, and our knowledge of quantum field theory and electrodynamics, we construct various symmetry requirements, and from that a Lagrangian, whose dynamical variables are phonons, representing the possible vibrations of the atoms, electrons, and photons. This allows us to deduce the Hamiltonian of the operator, and in particular the structure of energy bands for the electrons and phonons. From that, we can compute all the properties of lead. Its colour from the energy difference between the electron energy bands. Its compressibility and the propagation of sound waves from the phonon structure. Its brittleness from the electron energy bands at the boundary of the substance (what is the energy required to convert from a single crystal of lead into two crystals; i.e to reshape the energy bands in the associated way). It's melting point from the energy required to excite the valance electrons from the bands which hold the substance together. And so on. So in physics, we start with the energy band structure which arises from the effective Hamiltonian, and from that deduce the properties. The only observation we need (or would need, were we smart enough and good enough at the complex calculations) are the crystalline structure and electric charge and mass of the atomic nuclei -- not to feed into our theory as such, but to identify which of the various possible forms we ought to be focussing on. It is not these properties themselves that give us the information we need, but that we can feed them into a pre-existing method of abstraction.

But Locke asks us to start with the macroscopic properties, and from that specify the substance. But colour, texture, compressibility, and so on are not in themselves enough information to compute the whole energy bands of the metal. Locke's empiricist method is doomed to failure.

That's not to say that the empiricist method of induction is of no use whatsoever. People needed it to help work out the theory; but once we have the theory (not necessarily derived from the study of lead), we can apply it (and test it) in numerous other circumstances.

But there is a second, more subtle, problem with Locke's notion of substance. What he describes properly only relates to the form; that is to say that part of the substance which we can represent abstractly. That still leaves the matter (that part which we can't). Obviously, matter is inaccessible to Locke's method, which is probably why he ignores it. But substance is the union of form and matter. Our thoughts are, if correctly informed, an objective and accurate representation of aspects of an object, but they are not the same as the object. A bush doesn't sprout in our brains when we think of a bush. Therefore our thoughts can only represent part of the object. That leaves another part inaccessible to our understanding. We know it must be there, but it is impossible for us to understand in the same way that we understand form. Thus substance is the union between form and matter. But Locke's method, and the empiricist method in general, leaves out all notion of matter; not by reasoned argument, but due only to the empiricist presumption.

For example, do we understand triangles by combining simple properties. Some people might, but not everyone. I think of a triangle in terms of three points in a two dimensional Hilbert space: (a,b); (c,d); (e,f). Join those points together with lines which map out the shortest distance, and you get a triangle. Did the idea of space first come from sensation? Not necessarily. It is an abstract idea, and thus can arise from purely abstract thinking. Triangles exist in even the weirdest non-Euclidean geometries, but we are not capable of sensing them. We can still understand them. Triangles thus come first, and our sensations later. After all, we never observe the perfect triangle, just as we never observe the perfect straight line.

Thus it is not surprising that Locke finds the notion of substance vague and undefined. The true notion doesn't fit into his intellectual system. That's not the limitation of the notion of substance, but of Locke's system.

Truth and falsehood

Locke's presentation

Truth and falsehood belong to propositions rather than ideas. Ideas are simply appearances in the mind, and as such cannot simply in themselves be said to be true and false, although the word is sometimes used analogously to indicate the accuracy of the idea. However even such metaphysical senses of the word can be reduced to propositions. These senses are not of interest here. For truth or falsehood lying always in some affirmation or negation, mental or verbal, our ideas are not capable, any of them, of being false, till the mind passes some judgment on them; that is, affirms or denies something of them.

Ideas can be compared against something outside them, and then declared true or false in relation to their conformity to that thing. Some standard examples are: other men's ideas; real existence, and supposed real essences; all of which people compare their own ideas against. For example, we want our own notion of justice to be equal to the word as others use it. Our we might compare our ideas against some real existence. Thus ideas of men can be true, and of unicorns false; since men exist really, and unicorns don't. We can also reference our ideas against the real constitution and essence of something, on which its properties depend; almost all and possibly all of such ideas are false.

The natural tendency of the mind is towards knowledge. However, to gain knowledge of each particular one by one would be an impossible task. Instead, we make generalisation into bundles of particulars, and rank them into sorts, to which we give various names. If it gains knowledge of any of them, it can gain knowledge of any of that sort. When we encounter a new thing, we try to place it into one of our pre-existing categories, and thus by asking what it is we refer to the category by name, as though the name contained the essence of the species.

In this way men are predisposed to think that the abstract ideas in our minds must correspond to things external to that ideas. Simple ideas may be false, but are least likely to be so, because they are taken directly from sensual data. More complex ideas, however, particular mixed modes more than substances (which contain some reference to sensual data). It is not so easy to demonstrate that we use the words justice or cruelty in precisely the same way as other people. Our own notions might be false. Truth and falsity for things such as these can only mean that they are in agreement with other men. Abstract ideas of mixed modes originate entirely internally; we start with an idea (perhaps taken from sensual data), which is something internal to us. And then we extrapolate to get the mixed idea. There is thus no easy connection between this idea and the external world, and thus no way to compare between one man's ideas and another.

With regards to real existence, only our complex ideas of substances can be termed false. Simple ideas are extracted directly from nature. Their truth exists only in those appearances produced in us, and so determining the truth of an idea, we are basically comparing it against itself. We cannot know if one man's idea of blue is the same as another's, but don't harm ourselves by assuming that it is so.

Neither can complex ideas of modes be false, because no complex idea I have has any reference to anything outside it. However, complex ideas of substance might be false. Indeed it is evident that they are all false when looked upon as the representation of the unknown essences of things; so much so that nothing more needs to be said of it. Locke passes over that chimerical supposition, and considers ideas of substance as collections of simple ideas, taken from combinations of simple ideas existing together constantly in things, of which patterns they are supposed copies. And here they may be false, if they are put together in a way that has no resemblance to the external objects (such as a horse that barks). Or alternatively when all the right simple ideas of a substance are present, for example by making a brittle substance a bit too hard or a bit too little brittle. If one leaves out the brittleness from the idea of a substance, then it will not be false so much as incomplete.

Finding something true or false always requires a judgement of the mind. It depends on whether ideas or words are joined or separated. But ideas in themselves are neither true nor false. They are within the mind, and thus separate from the reality of things. However, it is a mistake to think that our ideas are the same as other people's, even if they are signified by the same name. Equally, it is wrong to judge an idea to agree with real existence, when it doesn't, or judged as being adequate when it isn't so. This might be something which unites a number of simple ideas which do really exist together, but leaves out those which are inseparable. The mistake is even greater if we think that a complex idea contains the real essence of any existing body, when it contains only a few of the properties that flow from a body. But in reality, the properties flowing from the basic essence of something are endless. Thus our ideas can never truly represent the real essence of something.

A man can only be aware of things outside him by way of the idea in his mind. These ideas in themselves are not false (unless they are mixed up from contradictory simpler ideas); for they represent nothing except the idea in the mind. But as soon as we try to say that the idea is in conformity with some real thing, we can make err; although the falsehood lies not in the idea but in the proposition that it conforms to the real object. Thus our own internal knowledge is always right; but it may be and usually is wrong to compare our ideas against the external world.

My comments

We can see here the influence that Locke's thought has had in the contemporary world. In particular, we can think of the idea of philosophical relativism (which should not be confused with physical relativity, which states the opposite), where "truth" about things is subjective. For if all we can know are our own internal thoughts and feelings, or what Locke calls ideas, and these ideas cannot a) be false and b) be compared against other people's views, then it is easy to see how the idea that there is no objective truth which all men can agree on, and we can only judge things by whether we think it true. Thus while everyone can agree that the sky is blue (albeit that I am writing this on a typically overcast spring day in England); we aren't in agreement with each other, since everybody has their own individual ideas about what is meant by "blue" and even what is meant by "sky". Thus when I say The sky is blue, and my friend says The sky is blue, we mean two different things; and if we mean two different things then we can't be in agreement. That the sky is blue thus can't be an objective truth; but merely a multitude of people each using the same words to express their own subjective truth. If Locke's reasoning is correct, then this is one obvious way in which his thought can be extended. And it would be a very dark path indeed; because unless truth is objective, then the concept is meaningless. We cannot communicate with each other. We cannot study nature -- all we can study are our own ideas. We cannot fall in love or help another person -- all we can do is fool in love with an idea, i.e. something inside us. But our idea is not the same as that person. We have learnt and can learn nothing about anything except our own fantasies.

But fortunately, Locke is wrong, in fundamental ways. His emphasis on properties -- sense data -- as the fundamental aspect of our knowing leads him to believe that we can only think of substances as bundles of observable properties. He knows nothing of the methods of theoretical science; particularly modern science and the importance of symmetry in defining our construction of reality. Through these means, what to Locke is chimerical becomes apparent to us: the true knowledge of the form of substances. It is important to note that Locke rejected the mathematical approach with a single word. He did not argue against it; he did not have any solid grounds for rejecting it. He just dismissed it out of hand.

When Locke states that truth and falsehood belong to propositions rather than ideas, he is setting up a false dichotomy. Locke treats matters of relation as ideas. Yet matters of relation, whether mathematical, logical, or physical (such as describing the spectrum of the hydrogen atom) can be propositions. Thus some ideas are propositions, and truth and falsehood in the proper sense can apply to these. There is a second, subtler, false dichotomy in Locke's statement. Truth and falsehood are not the only options. Some propositions are self-contradictory ("This is a lie"), and thus neither true nor false but nonsensical. Others are contingently true (if this is true, then that is true; or "it is now lunchtime"), and again by themselves neither true nor false.

Equally, mathematics allows us to express things precisely and objectively. We might disagree with the precise meaning of the word blue, and even if we did agree we would have no way of knowing that we were in agreement. Locke is right about that. But he is wrong to think that that is the only way we can describe things. We can define the measure of frequency in terms of some clear experimental data -- say the radiation emitted by transitions across the hyperfine splitting of Caesium-133. Once we have that (and everyone can agree on it), we can each specify wavelengths of light specifying each energy level in materials. When we see a blue object, we don't just have to use our eyes to analyse it. We can measure the wavelengths of light emitted from it, as a ratio of the Caesium-133 radiation, and calculate the relative intensity of each spectral line as measured in our own frame of reference. All that is objective (as we know how to convert from one reference frame to another), and we can be confident that everyone competent enough to perform the measurements will gain the same understanding. It is objective, and therefore we have a standard to measure against reality.

Neither do the concepts of truth and falsity of propositions wait for us to pass judgement on them and affirm or deny something of them. The truth (or lack of it) of the statement "2+2=5" does not depend on my judgement; it is objective. The truth of the statement "adult lions are bigger than adult mice" is similarly objective. Even if I were to judge that "2+2=5", it would not make the statement true. Rather, our judgements themselves can be true or false. The danger of Locke's presentation should be apparent. Locke makes our minds and prejudices, rather than reality, arbiters of the truth. It is a small step from this to say that nobody can be wrong. Yet it is obvious that a great many people are wrong about a great many things, and all of us are wrong about something. Once again, Locke might not have gone this far himself, but he paved the way for many of the dangerously nonsense opinions of the contemporary world.

But by combining theoretical analysis with hints from experiment, such as the relevant symmetries, we can figure out the effective Hamiltonian. And this tells us what we need to know about the substance's form, i.e. gain an understanding of the real essence of the subject. This is not easy, but we now know that it is certainly possible. All of these measurements can be done in an objective way. This gives us an objective standard against which to compare our ideas.

Locke is wrong because although he knows of experience, he does not know or appreciate experiment, and how it goes beyond basic experience. It allows us to frame our ideas in a way that Locke couldn't imagine. It provides an objective way about finding the truth of things.

Then we come to the question of generalisations. Once again, Locke cannot comprehend how the properly educated mind generalises. We do not make bundles of particulars, and rank them into sorts. We recognise instead that things exist in multiple possible states, and that these states have some things in common and others different. There is no single individual object; everything has many states, even the simplest beings. Thus while it is impossible for us to say that "this being is in that state" (because there is continual motion from one state to another, even as we say the words), we can accurately identify an object as being in one of a set of states; and by knowledge of the set we have knowledge of the object. But we can understand all those states together, using the mathematical representation. To Locke, then, we start with individuals, and put them arbitrarily into categories, matching each category on the basis of similarity of property. But in reality, there is no individual in the sense that Locke meant it; every being is in itself a category. The same process of linking one potentia with another to form the set of states that describes an individual being, by looking for possible continuous or near-continuous changes, can also be applied to map out the space of a species. Thus the process of placing objects into species is not as arbitrary as Locke makes it out to be. The essence of the category is not in the name, but in the form and matter; the set of states.

I would certainly admit that the abstract ideas I have represent external beings. But I have that confidence because I don't think as Locke does; I have access to a better means of understanding; namely the methods of mathematical physics in particular, and modern science in general. Thus our representations of nature, if properly formed, can truly represent the real essence of things. It is thus not a category mistake to compare them against nature. Rather, scientific study depends on such comparisons being possible, and we construct our abstractions so that they will be possible.

What I have suggested is that matters of physics are objective, and we can represent thing in such a way that our representation has a one to one relationship with reality. What of concepts such as justice and cruelty? Surely here, we can only come to the truth or falsity of such concepts by comparing with others? No. Firstly, it comes down to matters of definition. There is nothing wrong with me defining the word in one way, and someone else defining it in a slightly different way. It is not that one of us is right and the other wrong. Of course, if we wish to communicate with each other, then we need a dictionary so we can convert from one concept to another, but, with enough patience and going back to scientific and thus objective definitions, we can do so. Thus though my friend might regard a certain action as cruel or just while I do not does not mean that we could not both be correct, and indeed objectively correct. My friend might say that something is cruel using his definition, but also work out that under my definition it is not an act of cruelty. There is no contradiction there, and the two statements are simultaneously and objectively true.

But there is another sense in which concepts such as this can be objective. Ethics is (whatever people like Hume and Moore might have said based on their false premises) objective. To be good means that one's natural ends or final causes can be realised. Such ends are objective; they are part of the definition of a species (we start with the final causes, and map out the possible forms are consistent with those ends). For example, we are a species of rational, social animals. To be rational means to have the tendency towards grasping intellectual truths, which include sensual data but also the processes of abstraction. Similarly, an animal is defined as something which possess the animal and vegetative ends. That doesn't mean necessarily that those ends will be realised, or even that the being will be capable of realising them. For example, Kant, because he was befuddled by so many philosophical errors, certainly did not, and possibly could not, realise his intellectual ends. But those ends were still there, and because of that, Kant remains a rational being.

Thus goodness is objective, and by working through the details from those beginnings, we can come to objective definitions of virtues such as justice and vices such as cruelty. This doesn't mean, of course, that our own concept of justice (which will no doubt differ from the objective natural law virtue in various ways) is useless. But it does mean that the statement "justice is a virtue" is an objective truth (or falsity, depending on how closely our definition of the word justice lines up with the natural law definition) in the same sense that the statement "healthy grass is green" or "A hydrogen atom has such and such spectral lines" is objective. It is possible to define the concepts in a way that follows from the fundamental principles of nature, and once they are so defined then everything else follows.

Thus Locke is entirely wrong to say that such concepts are only internal to us, and cannot be compared between different people.

On Space, Time and Extension

Locke's presentation

We get the idea of space through our senses, sight and touch. We see a distance between bodies, or two parts of the same body, and from this derive the notion of both space and extension. Each different distance is a different modification of space; and each idea of a different distance is a simple mode of this idea. Measurements in inches, miles and so on, are so many distinct ideas made up of space. Once they exist in someone's mind they can repeat them without joining them to any body.

Another modification of the idea of space is the idea of figure, which is the arrangement of parts of terminated extensions. We get the idea of figure by adding together the various simpler ideas of basic extensions. There is an endless variety of different possible shapes; some of which are inspired by our senses, and others, once we know how to do it, we can piece together ourselves.

Another idea in this vein is place. We can consider the relation of distance between any two bodies or points. Any two points which are considered at keeping the same distance with each other can be thought of as being at rest. For example, if we have four points, and the distances between them are constant one day, we can consider them at rest; while if one of them moves we can say that it has changed its place. We can think of place relative to particular bodies; for example the movement of chessmen around the board is in relation to the chess board, and the fixed parts of the earth it rests on.

Since the notion of place can only be thought of in relative terms, we can't specify the place of the universe as a whole, although we can of all the parts of it. All beyond the particular beings is a uniform space or expansion, in which the mind finds no variety or marks. To say that the world is somewhere means only that it exists. The idea, therefore, of place we have by the same means that we get the idea of space, (whereof this is but a particular limited consideration,) viz, by our sight and touch; by either of which we receive into our minds the ideas of extension or distance.

Extension and body are not the same. Body is something solid and extended, extension the space that lies between the ends of those solid coherent parts. Solidity and extension are different ideas, even though the one depends on the others. For example motion (by which Locke means locomotion) cannot be conceived without space, and yet motion is not space, nor space motion, and they are distinct ideas. So we can suppose the ideas of space and solidity. And if spirit is different from body, because thinking lacks extension, the same reasoning would suggest that space is different from body. Extension includes no solidity, nor resistance to the motion of body, as body does. The parts of space are inseparable, both really and mentally, while we can divide up a solid body. The parts of space are also immovable. The definition of extension does not explain what a body is. The division of beings into bodies and spirits proves that space and shows that solid beings (extended and unthinking) and thinking beings (thinking and unextended) shows that space and body are not the same.

We get the idea of succession by reflecting on the appearing of various ideas one after another in our understanding. This is different from motion; every appearance of motion can be reduced to a succession of images. People don't observe motion unless that motion produces a constant train of successive ideas. We may constantly look at an object, and not observe it moving, while if we come back an hour later, we see that it has. However, even if things don't move, we still perceive a succession of images. From this succession, we get the idea of time.

My comments

There is one other type of space which Locke doesn't consider here, namely geometrical space. We can start with Euclid's axioms, for example (being simpler than hyperbolic, or Reimann's geometries), and from these come to an understanding about various mathematical and geometrical objects. Now Euclid's axioms (definition of line segments; definition of a straight line; definition of a circle; equivalence of right angles; parallel lines never meet) form a starting point from which practically every other geometrical notion (in Euclidean space) can be deduced. For example, one can deduce the existence of triangles, and the various rules governing their angles and side lengths, from these axioms and perhaps a few trigonometric definitions.

So where did Euclid get his axioms from? Locke might be right that they originally were inspired from sensual data. They certainly seem to be intuitively obvious; and the reason why that is so is that they seem to be in accordance with what we observe. However, from this point Locke moves onto slightly shakier ground. He is not a strong mathematician; and he is not truly aware of the mathematical process of deduction. But still, one might see how a modified version of Locke's thought might be consistent with this process. Note also for the moment we are discussing geometrical space; which to Locke is just an idea. Indeed, it should perhaps be seen as the exemplar of Locke's notion of ideas. We induce a few axioms from observation; and assuming that they are true deduce a whole mathematical system. But that system, it seems, only exists in our mind. We have taken something physical, and replaced it with something mental, and it is on that mental idea that we make all our conclusions. It is not clear, on this analysis, that those conclusions have any bearing towards the physical world. That initial step of induction is a problem.

Euclid's geometry stood for over two thousand years; until the great Carl Friedrich Gauss showed that it was not the only self-consistent system possible. His hyperbolic geometry tweaked Euclid's axioms, and worked out a new set of consequences. In particular his geometry was one in which parallel lines can meet. Later in the 19th century, Bernhard Reimann developed another self consistent geometrical space, one seeming moved even further from reality. This proves more problematic for Locke, because it seems that hyperbolic geometry was not derived from sensual data; and Reimann's geometry even less so. Essentially, these mathematicians just tried a different set of axioms to Euclid's, and came up with a different system. They certainly could not have got where they did without Euclid's original work, and Descartes' analytic geometry, or Newton and Leibniz's advances, but nonetheless we are getting removed from sensual data, and more along the lines of "let's invent some axioms, and see what the consequences are. One does not need sensual inspiration to do this; the only tools necessary are sufficient mathematical training and a strong imagination. But while they are still just mental ideas, and not seen as relating to anything physical, Locke can still remain happy.

But then, of course, the physicists have for some time been claiming that these are not just mental ideas, but do have real world significance. The process began, of course, with the medieval mathematicians, who proposed that physical space could be represented by an Euclidean geometrical space, and physical time represented by an Euclidean line; that is to say every conclusion drawn by the (Euclidean) geometers was also valid in the physical world. To be fair to Locke, by his time these mathematicians had not yet been proven right. Indeed they were only half-right: reality can be represented geometrically; but that geometry is not Euclidean. But in Locke's time, the thesis was certainly around; indeed some (such as Descartes) went further, and stated that physical objects were geometrical objects rather than the weaker but better statement that some aspects of physical objects are functionally equivalent to geometrical objects.

Einstein then took the additional step of merging the representation of space and time into a single geometry; firstly a hyperbolic geometry (special relativity) and then a Reimann geometry (general relativity). Quantum physics proceeds in a similar way; saying that other aspects of reality (such as the electromagnetic gauge) can also be represented mathematically. In other words, modern science is founded on the assumption that these abstract geometrical spaces are not just an idea in the mind, but an aspect of reality. This is a problem for Locke, and the empiricists in general. For the empiricists propose that we can abstract from nature (via our senses), but that we cannot in principle move back from our abstractions to say something about the physical world. If we take the perspective of the empiricists, modern theoretical physics should be impossible, or, at least, it cannot be connected in any but a very illusionary way to the real world.

So how do we know that the physicists rather than the empiricists got it right? Firstly, of course, there is the power of prediction. Physics assumes that there is some correct intelligible abstract representation of reality; we might not yet know what it is, but it is out there waiting for us to discover it. But if that is true, then it must be possible to perform calculations in our abstract space which will correspond to what simultaneously is happening in real space. We can then predict the state of the universe, and compare the abstraction against reality. Now, it might be claimed that this is no more than induction; we believe the theory because a great many of these predictions come true. But there is a bit more to it than that. We can apply the axiomatic method of the philosophers to physics. There is only a limited number of self-consistent possibilities, baring that we need to measure the values of various parameters, which are thus only known to a limited precision. Apart from that proviso, these possibilities are discrete rather than continuous. We can eliminate solutions which don't work, and just leave us with the one which does.

Which brings me to the second point: symmetry. We can observe symmetries both in nature and in the abstract world. Symmetries are something discrete; either a body has rotational (circular) symmetry or it doesn't. These are then used to construct the various possible theories. Violation of a symmetry usually leads to drastic consequences in the abstraction and thus the real world; in particle physics, for example, the number of types of basic compound particles depends on the symmetry group of the effective theory. Change the symmetry group, and you get a very different set of particles. Equally, we can't find a half-way house between Galilean space-time and a Minkowskian signature; it is either one or the other, and with the right experimental tools they are easy to distinguish. In that sense, it is obvious what symmetries apply. But moreover, some symmetries are necessary if physics is to work at all; the objectivity of nature leaves us no choice but to accept the symmetries that lead to Reimann geometry.

Is the concept of symmetry, as used in modern physics, an innate idea? I would propose that it is; perhaps not in the same sense that Locke's predecessors meant the term, but certainly it is not something derived from sensation, directly or indirectly. That is not to say that we can't see some symmetries in nature; I can look at an object that has approximate circular symmetry; and perhaps from that deduce the idea of symmetry under rotation. But this cannot be the case for the more abstract symmetries, such as the Lorentz symmetry group or SU(3) gauge symmetry. These we only discover by playing around with geometrical axioms. But once we do discover them and postulate their applicability to physics, then we quickly see their utility.

Locke's flaw in this matter, then, was his standard one. He described well one way of thinking about space and time; but did not consider that it might not be the only way of considering them. He did not appreciate the strength of the approach of the early mathematical physicists. He did not consider the possibility that geometry might not only be an abstraction induced from reality but a representation of reality. And with that, Locke's philosophy again falls short.

Substance

Locke's presentation

Locke didn't think that there was a clear distinct idea of substance. It doesn't help to feign a knowledge when we are ignorant. Names do not help us to understand what it is.

Various different meanings had been proposed. But there is a question whether it is applied in the same sense to the infinite incomprehensible God and to first spirits and then bodies. If it is in the same sense, then it would follow that God, spirits and bodies have the same common nature of substance, and only differ in a modification of that substance, just like a tree and pebble are both bodies and differ only in a modification of matter. On the other hand, if it applies in three different significations, then why use the same word to describe all of them? And if there are three ways of framing substance, why not also invent a fourth, or fifth?

The notion of accidents, a sort of real beings that need something to inhere in, forced people to invent the concept of substance to support them. The Indians could have saved themselves a lot of bother, instead of saying that the world was supported by an elephant and that by a tortoise, had just made all their problems go away through using the word substance. European philosophers do not know what substance is, only have a confused idea of what substance does, that it supports accidents.

What of our ideas of substance? Our minds are furnished with a great number of simple ideas. They note that these simple ideas are often associated with each other; and thus unite the ensemble in one name for quick reference. Not imagining how these simple ideas can subsist by themselves, we suppose that there must be some substratum in which they do exist, and which we therefore call substance.

So there can be no concept of substance except something which produces simple ideas in us. These qualities are called accidents. If anyone is asked where colour or weight resides, people would have to discuss solid extended objects; and if asked where solidity and extension resided in, then he would flounder like the Indian with his elephant. This talk is just like children, who, when questioned, give an answer that means they don't know what they are talking about. We then give the general name substance to this thing that that from which ideas are inspired subsists in.

We thus come to the ideas of particular sorts of substances by collecting such combinations of simple ideas that we frequently find co-joined together. It is the observable quantities that are put together to form the substance. However, our complex ideas of substances, besides all those simple ideas they are made up of, have always the confused idea of something to which they belong, and in which they subsist. Thus when we speak of any sort of substance, we say it is a thing having such or such qualities. Body is a thing that is extended, figured, and capable of motion; spirit, a thing capable of thinking; and so hardness, friability, and power to draw iron, we say, are qualities to be found in a magnet. These, and the like fashions of speaking, intimate that the substance is supposed always something besides the extension, figure, solidity, motion, thinking, or other observable ideas, though we know not what it is

We have no clear idea of what a substance is, whether a physical substance or a spiritual substance such as mind. We cannot see how the operations of the mind, thinking, reasoning, fearing and so on can subsist of themselves; so we invent the idea of spiritual substance to enable it. But we don't and can't know what this is. The idea of corporeal substance in matter is equally remote from our conceptions and apprehensions. The claim is then that we have no more right to deny the substance of spirit as we would affirm there is no body: since we are equally ignorant about both of them.

Whatever therefore be the secret abstract nature of substance in general, all the ideas we have of particular distinct sorts of substances are nothing but several combinations of simple ideas, coexisting in such, though unknown, cause of their union, as makes the whole subsist of itself. It is by such combinations of simple ideas, and nothing else, that we represent particular sorts of substances to ourselves. We invent species to convey the idea of those simple ideas that coexist together. They all adhere to that unknown commonality which is lacking in everything else. No-one can say that he has an idea of an object except in terms of those sensible qualities. Thus, the idea of the sun — what is it but an aggregate of those several simple ideas, bright, hot, roundish, having a constant regular motion, at a certain distance from us, and perhaps some other: as he who thinks and discourses of the sun has been more or less accurate in observing those sensible qualities, ideas, or properties, which are in that thing which he calls the sun.

Active and passive powers are a great part of the complex idea of substances. For as soon as people have an idea of any sort of substance, by gathering together the simple ideas which exist in it, among which its active powers and passive capacities can be considered. Thus, the power of drawing iron is one of the ideas of the complex one of that substance we call a magnet. Substances by these powers change some sensible qualities in other subjects; and thus we immediately recover the ideas of powers. Thus these powers are also part of what we think of when we think of substances.

My comments

Once again, Locke takes entirely the wrong approach to the subject. His fundamental problem is that he is trying to merge the word substance into an empiricist framework in order to understand it. Of course that isn't going to work. Substance is a concept in Aristotle's metaphysics (and other metaphysical systems, such as the neo-Platonic), which Locke rejects by adopting axioms which contradict it. For if all we can know about something is induced from sense data, then I can understand how the idea of substance can be obscure. But if, as I have suggested, we can also learn by reasoning, making predictions, and testing those predictions, then we can come to conclusions about the fundamental layer of reality.

One of the basic observations behind a sound metaphysics is that things change. Therefore they must be able to exist in numerous states; one of those states actual and the others potential. There is therefore no single way to describe any object; no single set of sensual data we might expect from it. Locke seems to be completely oblivious to this basic idea. Something must persist through this change, which is the substance or what is essential to the object. Other things might differ, which are the accidents. Now, obviously accidents is also used to describe things such as colour and texture. Something might appear blue in good light and have no colour at all in complete darkness; that depends on whether we define colour as a property of the energy bands (as I do -- in which case it would always be blue), or whether the object is physically reflecting certain wavelengths of light (in which case it has no colour in a dark room). Similarly, many textures and tastes change over time; so these would be considered accidental to the substance. But Locke takes this secondary sense of accident -- that of properties -- and makes it the primary sense. Now it is certainly true that the property of a being can't exist without the being itself existing. They need an underlying structure to subsist in. And it is also certainly true that our senses only have direct access to a being's properties, and not the underlying form or structure. If we accept the empiricist assumption that our knowledge can only come from our senses, then then it is easy to see Locke could come to his conclusion. But his conclusion rules out theoretical physics as much as it does metaphysics, so his assumption must be wrong.

The idea that a being can exist in multiple states is alien to Locke. He states that we observe that certain simple ideas -- basic sensual data -- often occur together, and so invent the concept of substance to explain their conjunction. But such an idea of substance is a singular thing; it would change as soon as our sensual data changes. Thus Locke's notion of substance is the very opposite of what the classical philosophers meant by the term. Equally, the classical philosophers worked from the bottom up; deriving the notion of substance from the unchanging underlying nature of beings, while Locke tries to define it in terms of the changeable external attributes. So what Locke calls an understanding of the substance is not what the classical philosophers meant by the term. To Locke, we have no hope of correctly understanding what classical philosophy was trying to understand, so we may as well not try.

It may be thought that Locke is just making an epistemological statement, about what we can know. But there is more to it than this. He is saying that substance is unknowable in principle. Substance is the union of form and prime matter, so he is saying that form is unknowable in principle. Now, we can divide the possible forms and substances into two classes: those we can in principle understand (not necessarily in practice at any given time, though as our science improves, so does our understanding), and those which we have no hope of understanding. Locke is saying that reality lies in the second class of beings. Thus, by his methodological assumptions, he is making a huge restriction on what reality could be.

Equally, the very process of converting sensual data into mental images carries with it certain assumptions. For example, when we look at something, we assume that the image is not distorted in normal circumstances (and we know about and can compensate for the exceptions, such as when an object is half submerged in a glass of water and it appears bent); we are assuming that light travels in straight lines and coherently through the air. This is not something we can directly observe; indeed any attempt to come to that conclusion through observation and induction alone would ultimately be circular. That is not to say that the assumption isn't right; but it is still an assumption. But that tells us something about the nature of light, namely an inherent tendency towards locomotion which doesn't noticeably deviate from a straight line.

Can we make a looser statement than Locke does? Does his conclusion show only that we can have no certain knowledge of substance? And there is an element of truth in this; we lack perfect knowledge even of the things we know well. But, when we come to consider the meaning of certainty and uncertainty we see that this looser statement does not help Locke. For being uncertain does not mean that we can have no knowledge, but only that we know that it is one of a limited number of options, and as our knowledge increases, that number decreases. Thus those who protest against the idea of the universe being indeterminate by saying that it means that there is a small but non-zero probability that they might suddenly jump into the garden and start to eat the slugs miss the point. It means that they might cook pasta that evening or they might cook rice, and that can't be predicted with certainty, but in either case slugs are definitely off the menu. Fortunately for ourselves, and even more fortunately the slugs. An uncertain knowledge can still be a very tightly constrained knowledge; very far from what Locke needs it to be for his enquiry to work.

So what about substances of God, of angels, and of physical beings? Are we right to use the same word to describe all three? Well, yes, if substance is the essence of the being; that which cannot change while it still remains the same type of being (we may distinguish between different types of being by considering the differences between continuous and discontinuous changes; not every change is achievable by a sequence of infinitesimal steps). This definition applies equally to God, angels and men, and thus is applicable without equivocation to all of them. Of course, the form of these beings, and (where applicable) the matter differs from one to the other, and from that we distinguish between the different types of substance. Locke again was confused. In one sense we already don't restrict ourselves to three different types of substance, but say that there are an infinite number of substances, one for each possible kind. In another sense, however, the notion of substance is united, for the word "substance" denotes the same thing in each different being.

Is substance just a name for that which we do not know? Again, historically, there might have been a kernel of truth in this. In the pre-scientific age, there was no accurate knowledge about the nature of water, its form of matter. People used the word substance without really knowing what made that substance differ from (say) vegetable oil. But they still knew, with certainty, that there had to be something there; something that remained constant through changes such as evaporation or freezing. Of course, that invites people to investigate the nature of the substances. That is to say their underlying form and matter. What it is about water that makes it different from oil? Of course, science has progressed a great deal since Locke's day, and we are now in a position to narrow down to the answer of those questions. We know (to a good precision) the effective Hamiltonians of water and glycerol; their energy bands; the possible vibrations and electron energy levels; how the individual molecules are bound as one and how they bind with other molecules. This is, in classical terms, knowledge of the form of the substance (or at least part of what goes into knowing the form), and thus at least partial knowledge of the substance. We can derive from the form the natural tendencies of the substances under various circumstances (such as when we heat glycerol in the presence of oxygen); and equally the various observable properties of these liquids such as their fluidity and colour. The medievals didn't know this; neither did the people of Locke's day. But the difference was that the medieval philosophy stated that such knowledge was there, waiting for us to discover it, while Locke's philosophy stated that all we could know about water is what we directly observe, its colour and fluidity and so on. Thus it is the empiricist philosophy of Locke, not the medieval philosophy (nor the mechanical philosophy) which is a hindrance to science.

Substance is not something obscure and ill defined. The notion of substance in general was clear to the medieval philosophers; as was the notion of form in general. What they lacked was how to apply that notion to particular substances; how to get from the general notion of form to an understanding of some particular form. This is perhaps what confused Locke. But Locke made the mistake of assuming that because it had not been done in his age, it could not be done and the whole enterprise to do it was a waste of time. The enterprise was begun by the Aristotelian philosophers of the late medieval period, and received fresh impetus starting from the mechanists of Locke's day. It has not been an easy task. But we are getting there, and in a few cases have very nearly got there. Just a pity that most scientists have forgotten the Aristotelian roots of what they are trying to achieve in their investigation into forms and substances.

Summary

This post has been rather too long, but has hopefully given an idea about what was wrong about Locke's thought, and, indeed, the notions of the empiricists in general. The aspects of Locke's thought I have been criticising here are the foundation on which the later empiricists built. Take away the foundation -- Locke's notions of ideas, and how they are related to sensation, how we build up complex ideas from simple ideas, and the futility of doing so if we are to try to understand nature at a level beyond mere physical properties -- and you take away the structure on which everything else was built.

But if you want the quick version, it is this. In the days when the first empiricist philosophers emerged, there were three known epistemologies. The first was that of the philosophers; who developed a metaphysics from general principles, and then tried to derive a theory of physics from those principles. The second was that of the mathematician, who had supposed that at least some aspects of reality could be represented by mathematical objects, and then investigated using the power of mathematics. The third was the empirical method.

Now in the days of the first empiricists, the philosophical method had stalled. Aristotle's had stalled because Aristotle had mixed incorrect specifics in with his largely (but not entirely) correct general principles. The mechanical philosophy had stalled, largely because it was waiting for Galileo and then Newton to breathe life into it. The mathematical approach had also stalled, partly because the key branch of mathematics needed to represent change, calculus, had not yet been discovered, and partly because the study of mathematics was undervalued during the Renaissance. The empirical method, however, was at that time making progress. However, it would have stalled, and reached its limits, because, as the likes of Locke and Hume so clearly articulated, it alone cannot give us the deeper understanding of the structure and substance of nature that we need. Furthermore, as Locke and Hume failed to understand, we need various assumptions in order to first process sensual data, in constructing our ideas (as Locke terms them) from the raw neural impulses. Our eyes, nerves and minds are (certainly in part, and most people would say in full) physical objects. To understand them, and what they give us and their limitations and where they go wrong, we need to be able to understand the physical processes that underlie them. And that understanding, or those assumptions, being prior to our sensation cannot come from our sensation.

The primary mistake that the empiricists made was to think that these three approaches were mutually exclusive. That you could only have one of them. But that is wrong. There is nothing contradictory in saying that some of our understanding comes from our senses, and some from constructing mathematical representations of reality, and some is deduced from metaphysical principles. The three then work together. We combine the three sources to obtain our abstract representations of reality, and then use the results to refine our philosophical and mathematical axioms, and to decide which new experiments to perform and how to interpret the results and improve their precision. This process of continual refinement, with all three sources contributing, and all three being improved, then continues until we finally reach the right answer.

This is science: not to trust in empirical data alone, or philosophical ideas alone, or in mathematics alone, but to use all three. It was the combination of them (underpinned by the mechanical philosophy) that led to the great advances in science up until the twentieth century. The philosophical leg was neglected in the twentieth century, as the mechanical world-view was shown to be incorrect. But replace it with the Aristotelian-Bayesian philosophy, and we have the recipe for scientific and social progress in the twenty first century.

Locke's main problem was that he lived before the quantum revolution in physics. Indeed, he lived before the Newtonian revolution. That doesn't by itself mean that he was wrong. But it is problematic for someone who was trying to argue that certain ways of understanding the universe were not possible. He lived far too early in mankind's intellectual journey to make that claim, and history has shown that we can know what Locke said that we couldn't know.



The failure of nominalism


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