# Mathematical reductionism: senseless

Does it make any sense to ask if logic can be reduced to math?

Truth be told I have no idea what the inverse logical reduction could look like. Naturally I'm familiar with a kind of reductionism, in scientific explanation. And I get the general motivation toward reductionism in philosophy of mind: matter is all there is.

• You have to understand logic before you can get any real insight into mathematics. If you were to write an introductory book on formal mathematics starting from scratch, you would have to start by introducing formal logic. – Dan Christensen Nov 18 '14 at 20:57

Yes, it does make sense. In topos theory, for example, different topoi can produce different logics, such that one can work in a topos without the Axiom of Choice, or without the Law of Excluded Middle.

In a sense, formal logic is arguably already a reduction of logic to mathematics. Despite seemingly being such a solid well-defined concept, mathematics doesn't have a single set definition that everyone agrees on, but in the sense of a rule-based, symbolic manipulation of complex concepts, modern formal logic very much belongs to the realm of modern mathematics.

The reverse project, of "reducing" mathematics to logic, was one of the chief goals of philosophers Bertrand Russell and Alfred North Whitehead, and the subject of their work Principia Mathematica. It was later shown by Kurt Godel, however, that not all of mathematics could be reduced to logic --no single system of rules and axioms can consistently produce all mathematical truths.

• I think your last statement is either a misunderstanding of Godel or a misunderstanding of mathematics. Modern mathematicians always try to prove everything from the axioms and do not consider something to be a theorem unless it is proved. What Godel proved, then, is not really that not all mathematics can be reduced to logic, but rather that any mathematical system is incomplete. – 6005 Nov 19 '14 at 4:25
• On second thought I think your answer is pretty good. I just wanted to point out your second paragraph is only one way of interpreting Godel's theorems. – 6005 Nov 19 '14 at 4:29
• @Goos Godel's work had a wealth of implications, but the one I highlighted is, I think, the one most applicable to the subject of the goal of uniting mathematics and logic. – Chris Sunami supports Monica Nov 19 '14 at 14:36
• People overstate Godel. His proof applies to those systems whose axioms take a certain form, which ensures they are safe from the manipulations that lead to Russel's paradox. It says nothing about more flexible systems. There could be a part of second order logic that avoided paradox in other ways. In other words, "all of mathematics" could be captured by limiting the boundaries around "mathematics" to something sane that includes all of the math we have already done. But all of the attention is focused on the fact that the most natural way fails. – user9166 Nov 19 '14 at 16:15

Classical logic is reducible to math (it can be mathematically described). The problem is math implicitely presupposes logic.

EDIT (precisions) it is possible to describe logic mathematically as rules operating on an abstract language (a set of symbols + a grammar for correctly formed sentences). This can be called a reduction of logic to mathematic. Mathematical reasonning, however, follows logical rules (set theory is expressed in a logical language) and the use of logical connectors is unavoidable. For example you'll need to say 'or' to define mathematically the disjunction operator (the logical or). So in place of reduction, we merely formalized logic inside a framework which is already based on logic. This formalization is not useless though: it serves meta-mathematical purposes, such as proving Gödel's theorem.

About reducing mathematics to logic: this project is known as logicism and was pursued by Frege Russell and others. Although large parts of mathematics have been formalised and we have a quasi-reduction, Russell's paradox proves it is impossible to have mathematics as only logic(+definitions of math symbols inside logic). You need specific axioms beyond logic to do mathematics (standardly, the axioms of set theory).

• Could you expand your answer elaborating on your views? – user132181 Nov 19 '14 at 14:44
• I edited my answer. – Quentin Ruyant Nov 19 '14 at 23:53