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I am aware of two major modes of reasoning used for justification of belief: deductive and inductive. Whereas physics relies on induction, mathematics seems to rely exclusively on deductive inference, which ensures the soundness of its conclusions. Is that correct? Are there other modes of inference used in mathematics?

Also, to be precise, what I am after here are the admissible modes of inference used for justification in the final product of mathematics, not the types of inferences mathematicians use in their daily work to generate mathematics.

For inductive reasoning, what I have in mind here is what is explaining in this entry of the SEP.

(if there is nothing in mathematics but deduction, mathematics is an extension of logic, and there is no other way to see the situation)

Frank
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  • What has your research uncovered so far? What exactly is the philosophical problem here? – Joseph Weissman Apr 05 '17 at 00:35
  • @JosephWeissman - just added a note: it's only deduction, therefore mathematics is "reducible" to logic, and there is no other way to understand mathematics - at least, that would be my position so far. – Frank Apr 05 '17 at 00:40
  • "mathematics is an extension of logic" -- Why bother to ask questions that you insist you already know the answer to? You've already made up your mind. – user4894 Apr 05 '17 at 01:08
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    You should also look at [abduction](https://en.wikipedia.org/wiki/Abductive_reasoning). Mathematicians use induction and abduction to generate and test conjectures, but when it comes to getting finished products (i.e. theorems) ideally only deduction from established results is accepted. Results that are not rigorously proved (by the standard of the time) are considered to be only preliminary. This does not necessarily turn mathematics under logic since it is using logic to do something else. Even if piano was the only instrument to play music music would not turn into piano mechanics – Conifold Apr 05 '17 at 01:45
  • mathematics most certainly does not depend exclusively on deductive reasoning! Principles of induction are central to mathematical teason // sorry, fumble-fingered. anyway, just try proving sth about natural numbers without using induction. betcha can't. –  Apr 05 '17 at 21:19
  • @mobileink - is that the same induction as we are talking about here? – Frank Apr 05 '17 at 22:57
  • Comments are not for extended discussion; this conversation has been [moved to chat](http://chat.stackexchange.com/rooms/56712/discussion-on-question-by-frank-what-inference-mechanisms-are-used-for-justifica). –  Apr 07 '17 at 11:27

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As a point of pedantry, mathematics doesn't use any mode of reasoning, mathematicians use modes of reasoning. And, naturally, like all people, they use a mixture of all different modes of reasoning to suit their needs.

However, we can talk about what sorts of modes of reasonings are depicted in the proofs that mathematicians provide to convey confidence in their ideas. In proof theory, we often see an operator known as "implies," typically represented by the turnstyle (⊢) which is a "metaoperator" in that no system defines the exact behavior of "implies." You're expected to understand what that means.

In a proof system, we set up a set of inferences which are deemed "acceptable for use" within the proof system in this way. For example, one might write {p, p→q}⊢q, which can be thought of informally to say "if p is true, and 'if p is true then q is true' then we can infer that q is also true." The list of these accepted inferences is typically provided up front. As a case study, one can look at First Order Logic, which defines a set of inferences in this way. Once the inference is accepted, it may be used in the proof from then on out.

So really, to answer your question, what we need to look at are the inference rules that mathematicians use. Many of these get phrased as axioms, so we can look there. If you look at the axioms we tend to use, most seem to be used to convey a deductive reasoning, but there are inductive examples as well. The one that comes to my mind is the Law of Induction, which is part of the Peano Axioms which define the natural numbers (0, 1, 2...). That law is very visibly designed to convey an inductive reasoning, to the point where they actually named the law "Induction!"

So mathematicians use both deduction and induction, along with whatever else is convenient to use. If you want to be more specific than that, simply look at the axioms which they use as part of their proofs, and you can see the reasoning they are trying to convey.

Cort Ammon
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  • I think we need to be careful about defining _induction_ properly, as it has IMHO a vastly different meaning in physics. To me, _inductive reasoning_ is the kind that is explained [here](https://plato.stanford.edu/entries/logic-inductive/#1) and that I think is not admissible as justification in mathematics. – Frank Apr 05 '17 at 23:09
  • @Frank I agree that they're definitely different, but I think there's a good reason the root words are the same. In mathematical induction, you prove a few statements which you assert are sufficient to prove what would otherwise take an infinite number of statements to prove, as you prove a property to be true for all numbers. I often like to quip that induction is the "1, 2, skip a few, 99, 100" approach to proofs. If you were a mathematician seeking to write down something you were using inductive reasoning for, mathematical induction would be the tool you'd use to pen it. – Cort Ammon Apr 05 '17 at 23:25
  • In other words, if I said "For every prime I know, I can find another prime that's larger than it, I believe that there's no largest prime," that statement would be inductive reasoning. You would then use mathematical induction to lock down the proof of that claim. – Cort Ammon Apr 05 '17 at 23:36
  • But two important characteristics of inductive reasoning (as I understand it so far) are: 1. it is _less certain_ than deduction ; 2. it relies on _observation_ of cases (in the natural world). Isn't mathematical induction more certain than this inductive reasoning? And also, it does not rely on observations of the natural world, right? – Frank Apr 05 '17 at 23:39
  • Inductive reasoning is observing a property in a subset of a population and making the assertion that that property will be observed in the entire population (contrasting with deductive reasoning which observes a property for an entire population and thus makes the assertion of that property for a specific subset). As for whether induction is more "certain," that is a question to ask in terms of proof theory. If you assume that you have a law of mathematical induction which is valid, then you can be quite certain of your inductive proofs. However, not all proof systems admit induction as – Cort Ammon Apr 05 '17 at 23:41
  • I'm wondering if mathematical induction is actually not _deductive_ in nature: suppose that property P is true for index n, and that I can deductively prove that if it is true for index n, it is true for index n+1, then property P has to be true for all n. – Frank Apr 05 '17 at 23:42
  • an axiom. As an example, Peano Axioms include a law of mathematical induction as an axiom because it was *not* proven that mathematical induction is valid and there were many properties that Peano wanted to prove which required it. – Cort Ammon Apr 05 '17 at 23:43
  • Inductive reasoning is usually thought to be less certain in the natural sciences because seeing a million white swans doesn't prove that a black swan won't be seen tomorrow, right? – Frank Apr 05 '17 at 23:43
  • Inductive reasoning is usually thought of to be less certain than deductive reasoning *in the presence of perfectly reliable axioms*. If you are 100% without-a-doubt positive that all men are mortal and that Socrates is a man, then you are also 100% positive that Socrates is mortal. In practice however, it can prove nightmarish to acquire one of those perfectly reliable axioms with which to do deduction. – Cort Ammon Apr 06 '17 at 00:13
  • Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/56623/discussion-between-cort-ammon-and-frank). – Cort Ammon Apr 06 '17 at 00:13
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The inductive/deductive division that you are pointing at derives from a division of thinking which takes something outside of itself to be explained (in physics, the world), and that which takes itself to be a self-sufficient world (in mathematics, the axiomatic).

These are rough demarcations as they cross-cross each other; for example there is an axiomatic presentation of QM, which given the demarcation above ought to be mathematics, but is in fact physics.

In both, imagination is important; and is in a way an irreducible remainder, meaning that it is this what remains after all methods are exhausted (see Feyarabends After Method).

Here's an important new example: infinity toposes interpret Martin-Lofs intuitionistic type theory (this means it doesn't support the law of the excluded middle); when one looks at the development of this notion, it is a tangle, a thicket, a forest of many different intuitions and notions which aren't readily shaken into the simple demarcation you're suggesting.

Mozibur Ullah
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  • Interesting. But for the axiomatic presentation of QM: at the end of the day "something outside of itself" is brought to bear, and can require a new theory. That's _induction_. As for infinity toposes, maybe the theory is not mature yet as mathematics, but the [first paper](https://arxiv.org/abs/math/0306109) I pulled on the topic on arxiv mentioned "definition" and "proof" - the author seems keen on integrating the edifice of deductively proved mathematics - he is not justifying via "intuition", apparently. – Frank Apr 06 '17 at 23:25
  • When I was taught QM, it was taught through examples, history and axiomatically; QFT however isn't taught axiomatically, but heuristically, for example Feynmans Path Integral; heuristics here, means physical thinking or intuition; in fact there is an axiomatic presentation of QFT - the Wightman axioms - but the only theory that can be proven to be built from that is just a *free* theory (ie a *non-interacting* theory, that is a theory 'free' from interactions). – Mozibur Ullah Apr 06 '17 at 23:54
  • The intuition for infinity categories is hidden away, one way to it is through what is called *enriched* category theory, when the hom-spaces are taken to be objects in a different category than the archetypal *Set*; in this case, they 'enrich' it in topological spaces. – Mozibur Ullah Apr 06 '17 at 23:57
  • Since the history of the subject is pretty entangled, I'm not surprised that the author doesn't attempt to go over it, or provide a synopsis; this might happen in the future when it is a mature theory. – Mozibur Ullah Apr 07 '17 at 00:04
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I am aware of two major modes of reasoning: deductive and inductive. Whereas physics relies on induction, mathematics seems to rely exclusively on deductive inference, which ensures the soundness of its conclusions. Is that correct? Are there other modes of inference used in mathematics?

Deduction is a way of arguing about ideas that has some limited applicability to specific kinds of problems. Induction of the sort that is supposed to be used in physics is nonsense:

Deduction vs Induction -- are they equally valid?

You want to know what sort of justification is possible in maths. The answer is that justification is impossible, for reasons explained here:

Do all epistemologies suffer from the "regress of justifications" problem?

Mathematical knowledge, like all other kinds of knowledge, is created by noticing problems, guessing solutions to those problems and then criticising the guesses. Guesses in some parts of maths, like parts of number theory, are fairly precise and axiomatised and in those cases, the results can be proposed and criticised using deduction. Other parts of maths are less precise, e.g. - applied maths used by physicists.

All maths is done using physical objects: pens, paper, computers, mathematicians' brains and that sort of thing. As such, our knowledge of maths is dependent on our knowledge of physics since we have to guess that these objects are modelling the maths we are interested in. Our knowledge of maths is restricted to a large extent by the laws of physics, see "The Fabric of Reality" by David Deutsch, chapters 6 and 10, and "The Beginning of Infinity" by Deutsch Chapter 8.

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alanf
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  • Can you explain how cohomology is dependent on or is restricted by, the laws of physics? – Frank Apr 05 '17 at 14:36
  • You think about cohomology with your brain, you write about it on paper etc. If your understanding of your brain, or paper or whatever is defective, then you may make mistakes about cohomology. So your knowledge of cohomology is dependent on your knowledge of physics. Note that this is true even if you think that there is something called cohomology itself that is totally independent of physics. – alanf Apr 05 '17 at 15:11
  • I see - so this is an extreme materialist point of view, where there is nothing but the physical world, so everything is physical, and therefore, ultimately regulated by the laws of physics? – Frank Apr 05 '17 at 15:21
  • The question is about how we learn about maths. Every means of learning about anything involves using physical objects that are regulated by the laws of physics. There are explanations other than those given by the laws of physics, e.g. - laws of biology, epistemology and so on, but they all have to be compatible with the laws of physics since they are instantiated in physical objects. If you want to call this materialist go ahead. If there is some sense in which pure cohomologies exist, or something like that, this has no bearing on how we create knowledge of cohomologies. – alanf Apr 05 '17 at 15:34
  • Wouldn't you say biology can be reduced to physical laws? What else is there? – Frank Apr 05 '17 at 16:08
  • @alanf "the question is about how we learn about maths". that is not the question you asked. learning math, doing math, convincing other mathematicians your work is good - 3 very different things. –  Apr 05 '17 at 21:42
  • sorry, i thought you were the OP. no, the question is not about learning. –  Apr 05 '17 at 21:43
  • @mobileink Doing math is learning about math. It's just learning math that nobody else knows. The laws of epistemology don't suddenly change when you happen to be learning stuff other people don't know. As for convincing other people, that's about them learning whether the content you produced is any good or not, and should be framed in terms of finding flaws in your idea, not convincing other people which is secondary at best, and can be destructive, e.g. - conforming to stupid fads, which happen in maths as in all other subjects. – alanf Apr 06 '17 at 07:16
  • @Frank The theory of evolution is not about some specific physical law, such as the equations of motion of quarks or electrons or whatever. A large part of biology is about the creation of structures with a high degree of adaptive complexity. Evolution can happen in any physical system with certain abstract properties involving storage of information and copying, variation and selection of that information. There is a chapter in "the beginning of infinity" about abstractions: you should read it. – alanf Apr 06 '17 at 07:27
  • @alanf - you avoided my question. I was asking what there could be in biology but the laws of physics :-) If there is some other kind of law that would not be reducible to physics. – Frank Apr 06 '17 at 15:48
  • The laws of physics alone don't prohibit things like spontaneous creation of adaptive complexity. The laws of biology and epistemology don't permit that. – alanf Apr 06 '17 at 15:59