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.

Filter by
Sorted by
Tagged with
1
vote
5answers
19 views

Why is there so little discussion / research on the philosophy of precision?

I was thinking the other day about the difference between rational and irrational numbers, and wondering whether the distinction between them is created by leaving out discussion of precision. So for ...
1
vote
0answers
32 views

Poincare says we are born geometric or arithmetic thinkers. Which was Grothendieck and why?

Poincare proclaims that the mathematical continuum originates from the sensible intuition and that intuition by pure number or logic alone could not have given us this notion. Intuition by pure ...
3
votes
2answers
66 views

Did physicist Max Born think that mathematical structures are platonic entities?

It seems that prominent physicist Max Born (https://en.wikipedia.org/wiki/Max_Born) believed in some kind of Platonism. We can infer this, for example, from the book "The Innermost Kernel" (https://...
1
vote
0answers
31 views

Is this an argument about the world or about human cognition?

This is a question about a thesis I have encountered regarding the relation of abstract mathematics ( Category Theory in particular ) with reality and the nature of human cognition. The argument goes ...
-2
votes
0answers
82 views

Do Gödel numbers can be used in order to logically determine the validity of Platonic Infinity? [closed]

"Historically, logic has been studied in philosophy (since ancient times) and mathematics (since the mid-19th century), and recently logic has been studied in cognitive science (encompasses computer ...
1
vote
0answers
70 views

Are the foundations of mathematics “doomed” to be set-theoretic in nature?

Let's say we want to come up with a foundational theory for all of mathematics and let's say that it is embedded in first-order logic. Note that the machinery of first-order logic is described with ...
2
votes
3answers
290 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 ...
-2
votes
3answers
350 views

Do Gödel's First Incompleteness Theorem imply the inconsistency of Platonic Infinity?

According to Modern Mathematics (where the majority of mathematicians agree about the notion of actual infinite sets, as established mostly by George Cantor) an inductive set (as given by ZF(C) Axiom ...
1
vote
1answer
226 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 ...
0
votes
2answers
187 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 ...
1
vote
3answers
64 views

Help with an existential natural deduction proof

From the assumption ∃x∃y R(x, y) I need to derive the conclusion ∃y∃x R(x, y) From the comments: I tried to use Existential Elimination but I can't figure out how to do it properly.
17
votes
11answers
8k views

Is Mathematics considered a science?

Science, generally is analyzing information gathered from observing phenomena, and coming up with theories to try and explain the phenomena. Then, attempting to predict a new phenomenon before it ...
0
votes
1answer
24 views

How does one go about this natural deduction proof?

From no assumptions derive the conclusion ∃x t = x (where t can be any term).
3
votes
5answers
3k views

Was Kant incorrect to assert all maths as 'a priori'?

Preface: Kant's assertion is rebutted by Prof David Joyce who references non-Euclidean geometry and by the last sentence on Sparknotes which states that 'empirical geometry is synthetic, but it is ...
3
votes
2answers
215 views

Does mathematical formalism have an opinion on semantics?

Mathematical formalism regards mathematics as a syntactic matter, where symbols are manipulated according to rules and the symbols need not have any meaning. I am wondering though whether it has ...
4
votes
1answer
66 views

Decidability of predicate logic

In the language of predicate logic with only identity and no predicates, function symbols, or constants, is it possible to construct infinitely many non-equivalent formulas?
6
votes
4answers
2k views

How long is the standard meter?

In the Philosophical Investigations §50, Wittgenstein writes: There is one thing of which one can say neither that it is one metre long, nor that it is not one metre long, and that is the ...
32
votes
17answers
13k views

How does mathematics work?

If I am given a parking lot with ten thousand cars and I want to determine whether one of the cars is orange, the only way I can do this is go through the parking lot examining each car until I find ...
3
votes
0answers
123 views

The notion of knowledge for Kant and mathematical objects

As far as I understand the notion of knowledge in Kantian philosophy, we cannot speak of knowing something unless there is a relation between its concept and some object of intuition showed in ...
3
votes
1answer
111 views

Did anyone argue against the possibility of a perfect prediction from within a system?

Did anyone offer an argument against the possibility of a perfect and complete prediction about a system from within that system along the following lines: Let's imagine a machine (like a desktop ...
1
vote
4answers
152 views

Argument against Platonism

Platonic view of mathematics states that numbers have abstract reality. One way to test what this really means is to do a thought experiment of extinction of humanity. Also suppose after all evidence ...
1
vote
3answers
99 views

Consistency of Axioms

In Godel's Proof by Nagel & Newmann, they write : In Riemannian geometry, for example, Euclid's parallel postulate is replaced by the assumption that through a given point outside a line no ...
-1
votes
3answers
225 views

Is Mathematics the Ultimate Culmination of Analytic Philosophy?

As a novice amateur, the similarities between mathematics and analytic philosophy seem striking to me. At least in a caricature view of analytic philosophy, it is the project of establishing the ...
0
votes
2answers
228 views

Existence of numbers on number line

Consider the fraction 10/3. One way to interpret this is by stating the following: in the process of long division, which is a rule, take divisor as 3, and take dividend as 10, and initiate the ...
-1
votes
3answers
155 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 ...
0
votes
1answer
118 views

Size of infinite sets [duplicate]

Cantor's method of comparing set size uses one to one correspondence i.e. existence of a bijection. Now, set A = (0, 1) and set B = (0, 2). Using the function x → 2 x, every element of set A can be ...
1
vote
1answer
127 views

Solipsistic Thinking :: Is there no maximum number;

I have something(s) that causes me both trouble with (keeping emotionally connected to what I'm trying to express) Λ (expressing myself). So please be nice and ask questions :: εὐχάριστος (Grateful) ...
2
votes
1answer
100 views

How do proponents of finitism respond to the claim that their position is “dubiously coherent”?

Michael Dummett writes (page 349) Since primality is decidable, the statement that any particular natural number is prime must be determinately either true or false, since the decision procedure, ...
-2
votes
2answers
262 views

Can incompleteness be eliminated by redefining the notion of a formal system? [closed]

The ONLY reason that we know that "cats are animals" and "2 + 3 = 5" are true is that these are defined to be true. The entire body of conceptual knowledge works this same way: Expressions of language ...
5
votes
3answers
305 views

Are the “laws” of deductive logic empirically verifiable?

"Is Logic Empirical?" strongly suggests a question that I would like very much to get a handle on. That phrase is a title of an article by Hilary Putnam, and, according to synopses/reviews, the ...
1
vote
3answers
255 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 ...
3
votes
0answers
95 views

Using differential equation to estimate epistemological growth constant

I found some tweets (1,2) describing a philosophy paper as follows: I came across this paper from the academic journal of philosophy that tries to solve a differential equation for an ...
2
votes
3answers
747 views

How is Wittgenstein’s “notorious paragraph” about the Gödel's Theorem not obviously correct?

Timm Lampert quotes from Wittgenstein's "notorious paragraph" (§8 of Remarks on the Foundations of Mathematics, Appendix 3) in http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6....
44
votes
15answers
9k views

Is Mathematics always correct?

It seems Mathematical theories/Laws/Formulas are the least questioned in all of the sciences. Is mathematics that good at being closest to the laws of universe, or is it just a logical tool of our own ...
2
votes
3answers
267 views

Understanding the simulation argument

I came across Nick Bostrom's paper called Are You Living in a Computer Simulation?. The paper argues that at least one of the following propositions is true: The human species is likely to go extinct ...
1
vote
0answers
157 views

Are Max Tegmark's Mathematical Universe Hypothesis and Seth Lloyd's Cosmological Model compatible?

I have been interested in Seth Lloyd's cosmological model (which proposes that the universe is a some kind of quantum computer or at least similar to it: https://en.wikipedia.org/wiki/...
2
votes
1answer
133 views

Did Whitehead express his motivation for writing with Russell the Principia Mathematica?

I imagine Bertrand Russell's motivation for participating in the project leading to the Principia Mathematica was an attempt to justify logicism and reject Kant's synthetic a priori, but what was ...
21
votes
4answers
34k views

What is the difference between a statement and a proposition?

I'm doing a MOOC on mathematical philosophy and the lecturer drew a distinction between a proposition and a statement. This is very puzzling to me. My background is in math and I regard those two ...
6
votes
2answers
717 views

What are the philosophical implications of using inconsistent mathematics?

Why mathematicians would prefer at times to work with inconsistent systems (from which I assume everything can be proven unless changing the logic used)? In particular, how could working with an ...
4
votes
1answer
101 views

Who first studied “logical (ir)reversibility”?

Who first studied "logical (ir)reversibility" philosophically? By "logical (ir)reversibility" I mean questions like:Why is it easier to multiply large numbers than to factorize them? understand a ...
21
votes
5answers
2k views

What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?

I have begun inquiring recently into mathematical aspects of Georg Cantor's theory of transfinite numbers and sets, which he developed between the years of 1874 and 1897. Throughout his theory, Cantor ...
2
votes
0answers
64 views

Transition of Mathematical Propositions

There are axioms, and then there are well established methods of working with them. It is clear that these methods are nothing but logical operations (rules of manipulation of symbols) on previous ...
-1
votes
1answer
162 views

Is the claim of mathematical objects being *abstract* necessary for understanding them and their applications? [closed]

One hears a lot about mathematical entities being abstract or at least spoken about as such. Now abstract is usually presented as non spatio-temporal, for example its said that number 2 is present [in ...
8
votes
3answers
685 views

Is Aristotle's resolution of Zeno's paradoxes vindicated by motion in the intuitionistic continuum?

In Physics VIII.8, Aristotle refers to his usual resolution of Zeno's paradox of motion: We should make the same response to anyone who uses Zeno's argument to ask whether it is always necessary to ...
0
votes
0answers
198 views

Is it there any theory or model in theoretical physics that is akin to Tegmark's Mathematical Universe Hypothesis?

Physicist Max Tegmark proposed a hypothesis that asserts that all mathematical structures do exist as universes. (https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis) But this hypothesis ...
10
votes
4answers
1k views

How do we separate rules of logic from non-logical constraints?

I think that very often the idea of 'constraint' appears in mathematics. For example, when a triangle is considered, 3 points are constrained not to be co-linear, and then we try to discover the ...
0
votes
0answers
186 views

Is it there any direct relation between Tegmark's Mathematical Universe Hypothesis and the Holographic Principle?

I would like to ask you about Tegmark's Mathematical Universe Hypothesis and its relation to the holographic principle: Could we use the holographic principle as a framework to Tegmark's MUH? I mean, ...
6
votes
1answer
2k views

What are the main issues on which the schools of Intuitionism, Formalism, and Logicism disagree?

What is the difference between Intuitionism, Formalism, and Logicism? Namely - on which issues do they disagree? And what is the relation of those schools of thought to Platonism, Nominalism, and ...
1
vote
1answer
178 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 ...
4
votes
4answers
281 views

Is there an alternative to Cantor's cardinalities that makes proper subsets smaller than their sets?

Cantor defined an infinite set as a set whose subset can be placed in a one-to-one correspondence with its subset. That is, take the set of all natural numbers: {0, 1, 2, 3, 4,...}. From that set, you ...