For want of a better word, mathematicism will be defined as the belief that everything is expressible in mathematical terms. I'm not sure if this is a position that anyone affirms, as my thoughts on the matter are nascent.

Anyways, I was wondering what sort of philosophical objections one could mount against specified mathematicism?


  • Just to be pedantic, how important is "expression" to your concerns? Are you particularly interested in your "mathematicism" as concerning the ability of scientific language to communicate through mathematical vocabulary? Is this a structural thing - do we want to say that there is a kind of isomorphic parallel or metaphor between what we can say in science and what we can say in mathematical discourse? Or are you suggesting something metaphysical - that scientific and mathematical words have a common underlying reference, the content of which is "math stuff"? – Paul Ross Mar 2 '15 at 11:33
  • Hi @PaulRoss, The latter (metaphysical "math stuff") is what I had in mind when submitting this question. – Five σ Mar 2 '15 at 11:37

I don't think that this is a view that's been explicitly explored in the contemporary literature. (User virmaior has pointed out that Descartes was responding to something like the view in question - see http://plato.stanford.edu/entries/descartes-physics/) However it does bear a resemblance to logical positivism. The positivists (at least some of them), were interested in constructing a language suitable for science. Specifically, they wanted to use a formal language, like that of Russell's Principia, to codify science. You might find something useful by looking them up! (I'm on mobile right now, but I'll see if I can dig up some references for you once I'm at a desktop).

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    Thanks! logical positivism is somewhat related to what I had in mind. – Five σ Feb 26 '15 at 20:00
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    Note that logical positivism also assume that you have a base vocabulary which corresponds to direct observation, so their position is: everything is expressible in logico-mathematical terms, on the basis if direct observation. If you meant pure math then it's more mathematical platonism (e.g.physicist Tegmark). – Quentin Ruyant Feb 27 '15 at 14:35
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    @quen_tin: right. If you're doing physics then your language will be FOL plus a stock of interpreted predicates corresponding to the 'axioms' for the kind of physics you're doing. – possibleWorld Feb 27 '15 at 17:35
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    I don't think this is correct: "The positivists (at least some of them), were interested in constructing a language suitable for science. Their language of choice was first order logic." Do you have a source for this? Carnap e.g. mainly used higher-order type theory (in Logical Syntax his Language I is a form of FOL, but he doesn't express any preference for it). – DBK Mar 1 '15 at 0:50
  • @DBK: you may be right. The FOL bit was off of the top of my head; I could be mistaken. I'll change the OP just to be safe. – possibleWorld Mar 1 '15 at 1:21

I am a little over my head here, but it may be in the works of Heidegger that the distinction between a thing and anything that attempts to represent it is called out.

Recalling from my reading, this includes the neurological impression a thing makes upon your sense of sight or touch or any other means of perception and the mysterious mechanism that transmits that impression to whatever your mind might be.

So certainly and obviously language, but I believe mathematics, too, can never be more than an imperfect representation of reality. However, it can be good enough: good enough for us to think meaningfully about it, to postulate theories on its hows and whys. So, this is no denigration of math, just a disquisition on epistemology - what it is possible for us to know.

I invite those with more developed educations to elaborate on this, and correct any errors.

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    It seems pretty spot on. It's worth pointing out that even such a basic thing as a proposition has to 'refer' to some actual thing for it to have a truth value. – Mozibur Ullah Mar 1 '15 at 14:25
  • The connection is indisputable, but the thing remains distinct from the reference, which is a thing in itself. – memphisslim Mar 1 '15 at 14:27

Logical positivism comes to mind as a theory that analyses propositions; propositions being those sentences that have truth values; and this by the correspondence theory of truth.

It's mainly associated with Russell and Wittgenstein.

If we take 'everything' to mean propositions in this sense; then it seems so. For example, one can trace the development of theorem-provers like Agda which are formalising statements in mathematics as one fruit of this early activity; another might be topoi as a language for physics, as outlined in NLab. Both developments rely on higher type theory which is an outgrowth of Russell's work.

However, Russell points out in his introduction that Wittgenstein had solved the problem he set out to solve, but 'the problems of life remain'; these are part of the things that can't be turned into propositions ie prominently ethics and metaphysics.


The straight answer is no. Kurt Gödel proved in 1931 that any formal system rich enough to describe the natural numbers cannot be both complete and consistent. Put in plain terms, what he was saying that any mathematical theory will either: - Contain statements that are true but cannot be proven, consistent but incomplete. - Contain statements that are proven to be true, but so are their opposites, complete but inconsistent.

This is not a philosophical objection, but a mathematical proof.

Closely related is the concept of undecidablility in computer science, that there are some problems that can never be solved in a finite amount of time by a computer. Since any mathematical statement should be computable, the answer again is no.

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    That's an argument against formal systems being able to express anything; not an argument against mathematics to express anything. Math is more than formal systems. That's what Godel showed. – user4894 Feb 26 '15 at 19:28
  • From what I understand of incompleteness, it's not really an argument against mathematics, though many interpret the theorems as such. – Five σ Feb 26 '15 at 19:45
  • "Math is more than formal systems." Can you please expand on that? Keep in mind that Gödel's theorem was a direct response to Bertrand Russell's Principia Mathematica, whose explicit objective was to construct all of mathematics from the bottoms up using formal logic. And Russell was considered to have achieved that objective until Gödel proved him wrong. – Alexander S King Feb 26 '15 at 20:26
  • "it's not really an argument against mathematics" you are correct, it isn't. It's just a formal statement of the limits of mathematics. – Alexander S King Feb 26 '15 at 20:32
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    Actually, the issue is not so clear, and Godel's theorems are more nuanced than you think. Some mathematicians interpret the theorems as saying only that we will need to postulate more and more axioms over time (also, these axioms must not be recursively enumerable by a turing machine...). But anyway, incompleteness is not a limit on what we can express, but rather on what we can prove, so all of this hardly seems very relevant to the question. – 6005 Mar 1 '15 at 10:32

Perhaps, by "everything", we mean everything measurable, or everything objective? As for subjectivity, it cannot even be methodically demonstrated to exist, let alone expressed in numbers.

Other Minds http://plato.stanford.edu/entries/other-minds/

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