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Please list both the problem/area and justify why it is important philosophically. This question doesn't cover questions that are only important within the philosophy of mathematics itself.

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What sorts of philosophical implications do you have in mind - philosophy of mathematics, or more general philosophy? – Carl Mummert Oct 5 '10 at 11:17
Do you mean e.g. Philosophy of mathematics? "constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. (...) some adherents of these schools reject non-constructive proofs, such as a proof by contradiction." – Américo Tavares Oct 5 '10 at 11:23
I show you an empty box. I then place two marbles in the box. Now I take one out of the box. Arithmetic implies something about how many marbles are in the box, on some level. It doesn't cause there to be one marble there (unless you have a wacky view on causation) but arithmetic can here be the basis for knowledge about the world. Much of physics works in a similar, if more complicated, way. – Seamus Oct 5 '10 at 13:18
"implication" is perfectly legitimate here. "President X's change of heart on Issue Y has implications for Project Z" is an acceptable use of "implication", despite there being no strictly deductive consequences involved. You're interpreting the word too narrowly. – Seamus Oct 5 '10 at 14:50
I won't vote to close, but I will say that in my opinion a SE-style Q&A site is designed to answer much more specific questions than this one, to which whole books are devoted to answering. – Pete L. Clark Oct 5 '10 at 22:01
up vote 23 down vote accepted

Wikipedia has a more detailed description for each one of them, therefore I will just list them and the main ideas.

Aumann's agreement theorem

Two people under certain prior conditions can not honestly disagree forever. In fact, Scott Aaronson have proved they don't have to exchange too much information to lead to an agreement. If the prior conditions are met and the disagreement lasts too long, then one side has to be dishonest.

Arrow's impossibility theorem

In short, there is no perfect voting system.

Free will theorem

Under certain assumptions, if we have free will, so does elementary particles.

Gödel's incompleteness theorems

There are statements in a sufficiently strong formal system that can't be proven true or false within the system. Some people use this to justify humans must be different from machines, since humans can prove theorems by using another formal system.

Tarski's undefinability theorem

Similar to the theorem above, it states truth in a sufficiently strong formal system can't be defined by that formal system. For people who believe people are machines, this implies people can't define truth.

The following theorems might be a stretch, but it looks like someone can use them in philosophical arguments.

CAP theorem

It shows there is no distributed system such that each machine store the same information, can operate while some machines are broken, and can operate even when some messages are lost.

Rice–Shapiro theorem

There is no algorithm to check if an infinite set have some non-trivial property.

Shannon's source coding theorem

The theorem states there is a hard bound on data compression.

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+1 for "The following theorems might be a stretch, but it looks like someone can use them in philosophical arguments." – Ravi Feb 22 '12 at 23:14
There should be a "favorite answer" option. :) – Sniper Clown May 11 '12 at 6:03

Goedel's incompleteness theorem is used by some (e.g. Roger Penrose) as part of a justification for why computers will never achieve consciousness.

The fact that all infinite dimensional separable Hilbert spaces are isomorphic has philosophical implications for the metaphysics of quantum mechanics.

Various results in dynamical systems theory related to chaotic systems limit what can be said about predictability and about what it means for a system to be deterministic. For example this paper by Ornstein and Weiss (warning: it's huge and will take a long time to download on slow connections) has been used to suggest that the distinction between deterministic and stochastic systems is flawed.

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There is a distinction between deterministic and stochastic systems, so whatever suggestion has been made is probably misguided. – Mariano Suárez-Alvarez Oct 5 '10 at 12:53
The fact that all separable infinite dimensional Hilbert spaces are isomorphic may have inspired claims in the metaphysics of quantum mechanics, but it surely does not have any consequences which are not purely mathematical (as Hilbert spaces only show up in a mathematical model of the physical world, it does not even have physical consequences!) – Mariano Suárez-Alvarez Oct 5 '10 at 12:54
I should have prefaced that sentence with something like "on the assumption that Hilbert spaces are a true (or correct, or accurate...) description of the world..." – Seamus Oct 5 '10 at 13:14
The Ornstein and Weiss paper shows that for any chaotic deterministic system of a certain sort, there is a stochastic system that is "alpha-congruent" to it. This alpha congruence has been interpreted (by Patrick Suppes, I think) as being roughly empirical indistinguishability. So the claim is that we can't empirically tell deterministic systems from stochastic ones. It's not an uncontroversial claim, but it is an "implication" that exists in the literature. – Seamus Oct 5 '10 at 13:16
I don't understand the distinction between "the distinction" and "the distinction we make". Who is doing the distinguishing in the first case? Obviously there is a formal difference between mathematical models that are deterministic and those that are stochastic. But that's not the point. – Seamus Oct 5 '10 at 14:10

Existence and uniqueness theorems for things like differential equations (the big canonical one being for ODE's) can be thought of as a philosophical foundation for a weak type of determinism.

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Game theory and fair division have implications of sorts for moral (and political) theory. The foundations of mathematics cut to the heart of logical possibility. Noam Chomsky sparked research into formal grammar, and while I don't have much knowledge of this area I believe it holds promising theory for work in the philosophy of language. The models of biological neural networks could hold implications for the philosophy of mind and consciousness. Computability theory is suggestive towards metaphysics (see e.g. Church-Turing thesis) and the philosophy of mind.

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Mathematical Realism is philosophically relevant and (imo) important.

If mathematical realism is plausible (and true) then many things about the natural reality can be infered by mathematical reasoning.

Conversely if mathematical realism holds many open mathematical problems might be resolved in un-expected ways and provide technological progress in important areas of life (or death, of course it is up to the use)

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But this question has infinitely many answers, as after deep thinking every principle can be applied universally. Application depends upon the thinker not the theory.

But recently I was reading the Hasse-Local-Global Principle. To speak naively ( for a rude version , ignore the things in the parenthesis ) it says that " a global solution can be obtained ( over $\mathbb{Q}$ ) if has local solutions everywhere ( on $p$-adics and reals ) " . So it speaks something about global influence based upon the local influence.

After thinking I have found two practical applications ( but there may be numerous ) :

  1. Suppose you want to develop a country ( Global level ) , then if each individual ( Local Level ) in the country changes himself, then the global change is achieved .
  2. Some individuals may have Global Goals and Local goals in life, Global goals like " I must be a doctor, I must be an engineer blah, blah .." , and local goals like " I must catch the engineering college bus, I must buy a pen to write exams " . So if an individual satisfy the local goals then automatically it implies the global goal.

So thats what I have found , and that made me wonder. Actually one can always find such things in every theory and principle.

  • First you need to consider the theory X, then strip of its context clothes ( I mean jargons that apply to 'local context, like Hasse-principle do says about degree, etc.. but I didn't mention it ) .
  • Then you need to search a day-to-day application, which matches the naive version, in some analogous manner .

I have still some theories that are universal . Let me edit and put it.

Thank you .

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