# Questions tagged [philosophy-of-mathematics]

Philosophy of mathematics asks questions about mathematical theories and practices. It can include questions about the nature or reality of numbers, the ground and limits of formal systems and the nature of the different mathematical disciplines.

780 questions
2answers
117 views

### Can zero be defined without some definition of one? Can one be defined without some definition of zero?

I would prefer to ask this in the math community, but that crowd is hostile toward anything hinting of philosophy. It is my contention that a construction of the real number system which begins with ...
1answer
123 views

### Are there recent coherence theory of truth for mathematical truths?

Are there any recent works (papers, books, etc) in philosophy of mathematics where it is given an account of mathematical truth in terms of a coherence theory of mathematical truth? I am interested ...
1answer
417 views

### Deductively sound formal proofs of mathematical logic? [closed]

How can this possibly fail to partition True(x) from Untrue(x) for every formal system? When we specify that True(x) is the consequences of the subset of the of conventional formal proofs of ...
1answer
41 views

### Can a different universe be built with three dimensions? [on hold]

Can you theoretically create a universe that will have the same dimensions but will look different? For example, a paper that has a limit, or another similar dimension can it be different?
1answer
254 views

### What's so bad about giving up the Axiom of Choice?

The Axiom of Choice (AoC) in set theory famously gives rise to controversial and counterintuitive theorems. (Examples: Banach-Tarski paradox and existence of non-measurable sets.) I'm aware of some ...
1answer
119 views

### Examples of theories that assume the existence of an “External Reality”?

In this paper written by physicist Max Tegmark (https://arxiv.org/pdf/0704.0646.pdf) it talks about "External Reality Hypothesis". Specifically, he says: Although many physicists subscribe to the ...
1answer
315 views

### What is the relevance of applicability to the natural sciences in pure mathematics?

I think I am coming to a good, new understanding of the relationship of pure mathematics to the natural sciences. A major concern of mine is just how reliable is rigorous (characteristically "pure") ...
0answers
67 views

### Is a “truly infinite reality” possible? [closed]

A "truly infinite reality" must follow these guidelines: While a "truly infinite reality" normally allows the "re-occurrence" of most entities in either exact or self-similar form (like for example ...
0answers
64 views

### Does Tegmark's hypothesis include dynamical mathematical structures?

Tegmark's hypothesis is the idea that mathematical structures are physical and thus have physical existence (https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis) Zuse's thesis says that ...
1answer
68 views

### Does Tegmark himself include paraconsistent mathematical structures in his mathematical multiverse hypothesis?

Tegmark postulates in his hypothesis that all possible mathematical structures would exist. But does he include also possible mathematical structures described by other types of logic like ...
0answers
113 views

### Relation of Mathematical Propositions to Natural Language

Treating Natural Language as a language game, what role does it play in our understanding of mathematics? Does natural language provide meaning to mathematics? Does a proof of a conjecture, say FLT,...
1answer
201 views

### Is there a natural example of a non-self-referential semantic paradox in philosophy?

A commonly studied paradox is the liar's paradox. The liar's paradox is to determine whether "this statement is false". The usual resolution is to state this the sentence is not actually a statement ...
2answers
26 views

### Help wanted - need descriptor for a partcular type/form of argument

I am writing a paper on cognition, and to simplify my discussion I need an adjective or descriptor for particular category of argument as follows: I am arguing for the necessity of a construct with a ...
3answers
264 views

### Can infinity be made finite in certain conditions?

In mathematics there are not only infinitely big numbers, but also infinitely small numbers. One can consider arbitrarily small numbers that can exist only in the mathematical world. For example, ten ...
1answer
66 views

### what is the ontology-ideology distinction in phil of math

Quine proposed a distinction between ontology - the doctrine of what there is - and ideology - the complex terms and predicates expressible in one's theoretical language (though in his 1971 paper he ...
3answers
121 views

### Why is 2 + 2 = 4? [closed]

It is clear that 2 + 2 = 4. It is also clear that applying the successor function on 1 yields the next number, i.e. 2, and this operation can be repeated infinitely. This method can be used to verify ...
3answers
129 views

### Can a problem be solved if there exists no solution for it in any context?

Is finding a solution for a problem in a given context is an attempt to find a solution for the problem in another context? For example, it seems some of the hardest problems of real analysis were ...
3answers
102 views

### Philosophy - If Space and Time are infinite and therefore infinite copies of us would end up existing, then wouldn't we still be gone after we die?

I have been pondering a question in my head. If Space and Time are infinite, then does that mean that Nietzsche's Eternal Return theory is true in the way that my life would recur, that when 'I' ('I' ...
2answers
77 views

### Syntactic VS Semantic Provability

Consider a new Conjecture C. The task is to determine whether the conjecture is true or false. Now let us suppose, after working very hard, we are finally able to establish the truth or falsehood of C....
3answers
177 views

### If we assume logic is correct, does it imply that our consciousness proccesses real information?

[major edits] Even if our consciousness is an illusion (even in the sense Denett suggests), the mere fact we see some information flowing across the universe means there is at least something that ...
2answers
85 views

### What's the meaning of this quote of Pythagoras on the good and bad principle?

Simone de Beauvoir attributed the following quote on the good and bad principles to Pythagoras in The Second Sex, page 114 : There is a good principle that created order, light, and man and a bad ...
1answer
79 views

### Correct Way of Handling A Corollary of A Corollary?

I have a conclusion S that is moderately interesting. While the corollary of S is more interesting, the corollary of the corollary of S is extremely interesting. Should I just label them corollary 1 ...
5answers
256 views

### Is getting 100 Heads in a row from a fair coin a miracle or not?

Suppose a man continues to toss a coin until he gets 100 heads in a row. Suppose the outcomes of all tosses from the 9999901th toss to the 10 millionth toss are all heads and 100 heads in a row didn't ...
3answers
105 views

### Probability calculus and Quantum Mechanics [closed]

I am not an expert and probably this question highlights this. Anyway, is the probability calculus used in Quantum Mechanics? Does the concept of probability adopted in Quantum Mechanics satisfy the ...
1answer
96 views

### Why is ZFC not as susceptible to Gödel's incompleteness as was the Principia Mathematica?

So, from what little I have read (such as this answer), it appears to be that one reason why the program of Logicism, as laid out in the Principia Mathematica, failed was that its goals (of finding a ...
0answers
22 views

### Discovery VS Invention in Mathematics [duplicate]

Asking specifically in the context of philosophy of Mathematics, on what basis do we classify or should classify a new expression as a Creation of Mind (Invention) OR a Discovery?
4answers
436 views

### Philosophy - Does Einstein's Block Universe theory prove Nietzsche's Eternal Return theory is true?

If the Past, Present, and Future all exist in exactly the same way, then every single moment would be a ‘Now’ moment for me. it would also mean that me being dead in the future is equally real in the ...
1answer
88 views

### Could generalization of scientific theories be possible by just adding an ad hoc hypothesis?

In a seventeenth century world the Newtonian model did mostly very well to describe how gravity works in the universe and did well with most empirical evidence of that time. Of course now we know that ...
9answers
1k views

### Is a proof still valid if only the author understands it?

Some time ago I was reading about the recent Shinichi Mochizuki's proof for the famous ABC conjecture. It's enormous and so incredibly difficult that at that time virtually nobody was able to ...
2answers
211 views

### For a mathematical realist, is there a distinction between real mathematical objects and constructed mathematical objects?

Mathematical realists believe that mathematical entities exit independently of human minds. Mathematical objects have an objective independent existence, and they are discovered by mathematicians, not ...
25answers
7k views

### Why is there something instead of nothing?

A simple but fundamental question. The "something" means the whole Universe (known and unknown), it could be represented as the reality version of the set of all sets, which is itself debated. It ...
1answer
325 views

### Was Kant an Intuitionist about mathematical objects?

In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-...
3answers
677 views

### What are the discoveries that have been possible with the rejection of positivism?

I am wondering if the rejection of the positivism movement in philosophy lead to any major discoveries in mathematics and natural sciences? I am thinking it might have been able to contribute to those ...
4answers
2k views

### What does this Jacques Hadamard quote mean?

What does this Jacques Hadamard quote mean? The shortest path between two truths in the real domain passes through the complex domain. Is this a philosophical statement? what is its mathematical ...
1answer
252 views

### Is there any physics-model version of Tegmark's hypothesis?

Tegmark's mathematical universe hypothesis is very interesting (https://en.m.wikipedia.org/wiki/Mathematical_universe_hypothesis) but it has virtually no support among physicists because it is too ...
0answers
67 views

### The nature of Nominalist Formalism

In this entry in the stanford encyclopedia of philosophy, it is stated that the theory of nominalist formalism deals with the metatheory problem of formalism as follows: Commendably, Goodman and ...
2answers
251 views

### can we reason about logic?

People who study mathematical logic make arguments about logic itself. So it seems that people take for granted an "intuitive logic" (otherwise, how would they form arguments?). So the observation is ...
3answers
198 views

### Mathematical Consensus

Can anyone give me a reason why mathematics may require consensus to determine the quality of knowledge from the general mathematical community? Also what would be the counter to such a claim, as in ...
1answer
209 views

### Are mathematical results influenced by the way we reason?

Intuitions of mathematicians, and the mathematics they develop, are ostensibly influenced by whether they primarily rely on visual_spatial and/or verbal_symbolic reasoning skills. Is it fair to say ...
2answers
168 views

### Where can I learn about the philosophy behind mathematical and logical proofs?

I'm looking for something that dives into the philosophical idea of a "proof," and explains how the subjects of mathematics and logic deal with it. Does anyone have any book or article recommendations ...
2answers
199 views

### Does philosophy of mathematics affect mathematical research?

I am interested in a special case of the general question about whether the philosophy of X has an effect on the research or practice of X. My special interest is in the area of mathematics. I am a ...
1answer
169 views

### Where to start with the philosophy of mathematics?

This may be a duplicate question. What is the best way to get started with the philosophy of mathematics? Given that I know (from university) the basics that are discussed (Set theory, Russell's ...
1answer
120 views

### Is mathematics something real or just an abstraction we created?

Is mathematics something real like something so correlated in our universe which would be different in other theoretically universes or is it just an abstract universe-independent layer/framework we ...
2answers
81 views

### Must the physical phenomenon of the universe be differentiable?

The use of Calculus for the analysis of real-world phenomenon depends entirely on our universe not only being continuous, but being differentiable. By "real-world phenomenon" I mean things like the ...
1answer
114 views

### Can you imagine a completely different logical/mathematical system than that we have?

Can you imagine a different logic and mathematics? For example, with a different arithmetic, or even a universe with no logic or mathematics and contradictions? A non consistent system?...
6answers
7k views

### Falsification in Math vs Science

In the beginning it was thought that the statement 1+1=0 is false, and necessarily so. However, with the birth of modular arithmetic, it was found that indeed, 1+1 does indeed equal to 0 (in the mod 2 ...
0answers
69 views

### What was Wittgenstein's argument against Cantor's transfinite numbers and where did he make his objection?

G. E. M. Anscombe had this to say about propositions in Wittgenstein's Tractatus: (page 137) It seems likely enough, indeed, that Wittgenstein objected to Cantor's result even at this date, and ...
1answer
201 views

### Gödel's Results and Philosophy of Mathematics [closed]

Gödel's results essentially conclude that there are True but Unprovable statements in arithmetic. My thoughts are as follows: Axioms form the foundation of mathematics -because we need to assume ...
2answers
213 views

### What does Wittgenstein mean when he says “there are no numbers in logic”?

From the Tractatus: 5.453 All numbers in logic stand in need of justification. Or rather, it must become evident that there are no numbers in logic. There are no pre-eminent numbers. What does ...
1answer
196 views

### Are axioms in mathematics comparable to hypotheses in experimental sciences?

Remark: my question deals more particularly with the axioms of set theory, arithmetic, probability theory, etc. I think the status of the axioms in geometry is clearer. The French fictitious ...