What's the matter?

Matter is the stuff of physical reality.

Ordinary material things seem to be composed of atoms. But are we composed of nothing but atoms? We certainly interact with the world around us, and in particular, we can see other people in the world around us, so we are clearly part of physical reality. And as well as being physical, we clearly have feelings. So, physical things can have feelings. But how could physical things have feelings if physical things are composed of nothing but physical particles?

How could any amount of atoms possibly add up to something with feelings?

Does the difficulty of seeing how that is possible mean that we are not physical? Well, it would make as much sense to conclude that we do not have feelings. Does the difficulty of seeing how some huge number of atoms could possibly add up to something with feelings make it rational for us to doubt the atomic theory of matter? Well, there is a vast amount of scientific evidence for the atomic theory of matter. Still, perhaps that difficulty does make it rational for us to doubt that people are just a lot of atoms. Perhaps it makes it rational for us to doubt materialism. Does it make it rational for us to doubt the logical possibility of materialism, though? That is unlikely in light of how little is known about what matter is.

If materialism is true, then matter is such that some structures that certainly seem to be composed of atoms do have feelings, and in order show that such material structures could not possibly have feelings we would need to know a lot about the nature of matter, and we actually know very little about what matter is, beyond the utility of modelling physical things as being composed of particles that cannot go faster than the speed of light, and what it feels like to be physical.

In short, we cannot show that materialism is contradictory; and it makes sense that if a hypothesis is not contradictory then it is, in all likelihood, a logical possibility. That is because when we say that something is not a certain way, we mean that it is not the case that it is that way. That means that it cannot be the case that something is a certain way and that it is not that way. So, a contradictory hypothesis is not a logical possibility. And for what other reason could we rule a hypothesis out on logical grounds? It is far from obvious that there could be another reason.

Now, it is precisely because we do not know much about what matter is that matter could, for all we know, be something that was deliberately created, by some non-physical creator, out of nothing but the logical possibility of matter.

Note that the possibility of there being such a creator is not the same as the possibility that the picture of a God that atheists pick up from the religions around them is a realistic picture. It is irrelevant, here, how unrealistic such a God may seem; what matters here is that logically, the concept of such a creator has never been shown to be contradictory (despite the best efforts of the brightest atheists). So, rational atheists should have as little problem with the logical possibility of theism as they do with the logical possibility of materialism.

However, to remain atheists they must, in light of my proof that atheism is illogical, either show that the logic of my proof is flawed, or deny one of its premises. And insofar as "logic" does not mean formal logic, there is nothing wrong with the logic of my proof. And rational atheists could hardly deny the existence of things like these words and remain rational, so they would have to deny the other premise of my proof, the logical possibility of theism.

Or would they? Could they not say that "logic" does mean formal logic when extraordinary rigor is required? Extraordinary claims do require extraordinary justification, and formal logic is as rigorously logical as algebra. Still, formal logic is as rigorously logical as algebra because formal logics are algebraic structures. Formalizing a logical argument makes a mathematical model of the argument. It gives us a rigorously logical piece of mathematics, in addition to the original logical argument. And redefining "logic" would not make that argument less logical.

If I defined "matter" so that atoms could not possibly add up to something with feelings, would I have shown materialism to be false, or would I have given "matter" another meaning, different to the meaning that it has in any reasonable definition of "materialism"?

D E F I N I T I O N S

Leibniz was caught up in fierce philosophical disputes in the seventeenth century, but he hoped to bring peace to philosophy by inventing a logical calculus. Given a logical calculus, disputing philosophers would be able to sit down and calculate which of their views was correct, or so Leibniz believed. By the end of the century, he had built a mechanical calculator capable to adding, subtracting, multiplying and dividing numbers up to a hundred million, he had invented the infinitesimal calculus (following Newton's invention of the method of fluxions), and he was disputing Locke's view of the counting numbers (1, 2, 3, and so forth). Insofar as the counting numbers are composed of ones (1, 1 + 1, 1 + 1 + 1, and so forth), the fundamental arithmetical concepts are unity (1) and addition of unity (+ 1), and Locke thought that we acquired those concepts by observing a world of things. Leibniz realized that in order to see things as things we must already have got the concept of an individual thing. And by expressing the logical axiom that things are identical to themselves in symbols as A = A, Leibniz could get from that algebraic equation to any arithmetical equation by taking logical steps. For example:

Replacing A with 2 + 2 gives us 2 + 2 = 2 + 2, and in three steps, each of which uses one of three definitions (2 = 1 + 1, 3 = 2 + 1, and 4 = 3 + 1) with the logical rule that replacing things with equal things preserves equality, we get 2 + 2 = 4 like this:

          2 + 2  =   2 3 + 1 +2
          2 + 2  =   2 3 + 11 + 1
          2 + 2  =     2  3 1 1+ 1
          2 + 2  =   2 3 + 1 +4

With the logical steps of that proof of 2 + 2 = 4 laid out so clearly, it is easy to see why arithmetic might be nothing but logic and definitions, and why 1 and + 1 do not need to be defined in order for such a proof to be perfectly logical. Leibniz wanted to bring such clarity to all of our reasoning, by expressing in symbols all of the logical axioms and rules of inference used in logical thinking; and by assigning a unique prime number to each elementary concept, so that the logical relationships between ideas composed of such concepts could be calculated mathematically, although Leibniz died before he could complete that calculus.


Euler proved over a hundred theorems about the counting numbers in the eighteenth century. He also conjectured that if we multiplied a counting number bigger than one by itself five times, then in order to express that fifth power of that number as the sum of other counting numbers raised to the fifth power, we would need at least five of those other numbers. A couple of centuries later, a counter-example to that conjecture was discovered: 144 only needs four other numbers. Now, it makes sense that in order for that number to have been a counter-example to Euler's conjecture, that number had to have been a number of the kind that Euler was writing about. And for Euler, 144 was 144 ones added together. Euler thought that 1 + 1 = 2 by definition of 2. And one thing and another thing are two things by definition of two. But twentieth-century mathematicians invented infinitely many algebraic entities in order to build arithmetic out of them. Could the meaning of 144 being 144 ones added together have been 144 invented entities being collectively called "144"? Could a rigorously logical proof of 2 + 2 = 4 be based on, say, fictional dots?

Well, the reason why 2 + 2 = 4 that, firstly, 2 + 2 = (1 + 1) + (1 + 1) because 2 = 1 + 1 by definition of two, and secondly, (1 + 1) + (1 + 1) = 1 + 1 + 1 + 1 because how many things we have does not depend upon which ones we look at first, and thirdly, 1 + 1 + 1 + 1 = 4 by definition of four.

The English name for how many things there are when there is one thing of a certain kind and another thing of that kind and another thing of that kind and another thing of that kind and no other things of that kind is four, and "1 + 1 + 1 + 1 = 4" is just that dictionary definition of four written in quotidian symbols. Dictionary definitions describe the meanings of such basic words, they are not how such words get their meanings; words like four and two get their quotidian meanings in much the same way as words like pound and and get theirs (a way with which we are all familiar), and while symbols can be given algebraic meanings with algebraic definitions and axioms, quotidian symbols like 4 and 2 get their quotidian meanings in much the same way as £ and & gets theirs. So, is there any room in that reason why 2 + 2 = 4 for fictional dots?


Frege completed a logical calculus in the nineteenth century. And in order to show that arithmetic is nothing but logic and definitions, he tried to build arithmetic out of logical axioms and definitions, using his logical calculus. He added to Leibniz's definitions by defining 1 to be 0 + 1, where 0 and + 1 were defined in terms of concepts that certainly looked very logical. That meant that Frege could prove that 1 + 1 = 2, in what he took to be a rigorously logical way. Frege also noticed that Leibniz had not shown that 2 + (1 + 1) = (2 + 1) + 1, in his proof of 2 + 2 = 4. Now, it is obvious that how many things we have does not depend upon which ones we look at first. And if we had to prove every obvious step in a proof, in order for it to be a logical proof, then we would never get to prove anything logically.

Logical proofs begin with descriptions that are obviously true (or descriptions that would clearly be true were the assumptions of the proof true) and then, step by logical step (each step being such that from something true it would clearly take us to something true), they take us to conclusions whose meaning and truth are thereby obvious.

However, Frege thought that logical proofs should be more like mathematical calculations. Mathematical calculations are rigorously logical. Does the logic of a rigorously logical proof have to be axiomatic, though? Deductions from axioms are not always as logical as they look. For example, the axiom that a precise description of reality is either true or else not true is assumed by many rigorously logical proofs; but it follows from that axiom that "this uniform dot is red" is either true or else not true, and the dot in question might be such that the human eye could not possibly tell if it was red or not (that is because if there were no such dots then there would be a sharp line in the spectrum between the part that is red and the part that is not red, and sufficiently small dots next to such a line would, on both sides of the line, look the same to a human eye), and it is the human eye that defines what red is. And after all, logicians hypothesize logical axioms when they have lots of similar proofs, all of which are clearly logical. So, logical axioms are bound to be less certain than lots of logical proofs, and so rigorously logical proofs do not have to be deductions from axioms (logic and proof are all about certainty).


Russell discovered a logical paradox at the start of the twentieth century (following Cantor's discovery of a mathematical paradox at the end of the nineteenth century), effectively ending Frege's attempt to show that arithmetic is nothing but logic and definitions. So, Russell designed his own logical calculus. And after hundreds of pages of little but algebra, he obtained 1 + 1 = 2 where those symbols got their meanings in a rather algebraic way from such axiomatic assertions as that there are infinitely many things (written in symbols). Could Russell have shown that arithmetic is nothing but logic and definitions by reaching such equations after so much algebra, though?

The English name for how many things there are when there is one thing (of a certain kind) and another thing (of that kind) and no other things (of that kind) is two, and such dictionary definitions are what those learning the meaning of "arithmetic" take arithmetical truths like 1 + 1 = 2 to mean. So, the reason why 1 + 1 = 2 is a logical truth is that those things can be any logically possible things.

And the reason why every arithmetical truth is a logical truth is that there are infinitely many logically possible things. However, for Russell's rather algebraic equation to be an arithmetical truth, "arithmetic" would need to be redefined. And for it to be a logical truth, "logic" would have to be redefined, and we would need a very good reason to redefine "logic" (reasoning is logical when it is bound to take truths to truths, and we cannot pursue the truth by redefining "truth"). Was Russell's paradox a good enough reason? Every logical puzzle has a logical solution (because logical reasoning will not take truths to falsehoods), but mathematicians do like certainty, so they did not like having an unsolved puzzle at the heart of the logic of arithmetic. Redefining "arithmetic" so that its logic became the logic of algebra made a lot of sense to them. So note that there is nothing dubious about the logic of arithmetic. If I have a crimson object and I conclude that I have a red object, that would clearly be a logical deduction, and if I also have a scarlet object and I conclude that I have at least two red objects, would that not be similarly logical? And what if I have no other red objects and I define 2 to be the number of red objects that I have; would I not be giving "2" another meaning in addition to its arithmetical meaning?


Hilbert turned Frege's logical calculus into the mathematical model of simple logical reasoning that became the standard logic in mathematics and philosophy. And with Hilbert's encouragement, mathematicians defined 0 and + 1 within an axiomatic set theory, in order to avoid paradoxes like Russell's (and Cantor's). With those definitions, mathematicians can prove in a rigorously logical way that 2 + 2 = 4 where "2" and "4" are names of axiomatic sets. And that is the standard meaning of "2" and "4" in academic arithmetic. Mathematicians do like certainty, and because axiomatic sets have only those properties that their axioms describe in symbols, set-theoretic arithmetic is as rigorously logical as algebra. Nevertheless, axiomatic sets are algebraic models of collections whose sizes are numbers composed of ones, and because such models are bound to be a little inaccurate in some rather obscure ways (because of paradoxes like Russell's), the steps that would be needed to take us from that rather algebraic proof of 2 + 2 = 4 (where "2" and "4" are names of axiomatic sets) to a logical proof of 2 + 2 = 4 (where those symbols have their quotidian meanings) could not all be as rigorously logical as the steps of an algebraic proof. What, then, would a rigorously logical proof of 2 + 2 = 4 look like?

Well, the fact that two things and another two things make four things can be seen by counting them: there is (to begin with the first two things) a first thing and a second thing, and then there is a third thing (the first of the other two things) and a fourth thing (the second of the other two things).

It is obviously the case that counting a small number of things can tell us how many of those things there are, and the quotidian meaning of "2 + 2 = 4" is that two things and another two things make four things, so that was a logical proof of 2 + 2 = 4. Perhaps that proof would be more rigorously logical if it included an explanation of how counting works. But would we get a more rigorously logical proof of 2 + 2 = 4 if we used the standard logic and set theory to model that proof? The resulting model would be a rigorously logical proof (it would be a rather algebraic proof of an algebraic model of 2 + 2 = 4, and algebra is rigorously logical), but would it be a rigorously logical proof of 2 + 2 = 4?