Questions tagged [foundations-of-mathematics]

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Why do mathematical platonists believe in the abstract when math clearly comes from FOL, a non-abstract?

To assure ourselves first order logic is as free of paradox, errors, and impermanence, mathematicians and logicians "grounded" math in a language/system everyone can agree upon. Here is a ...
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How can you analogize mathematical induction to dominoes falling, if some domino can fail to topple?

This analogy doesn't convince me, because what if some domino (after b, the base case) fails to topple? In real life, a domino can remain standing upright if it got placed too far apart from the ...
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Is Mathematical Platonism meaningful to even speak about? [closed]

Certain propositions can be meaningless. How do we know if "Are there abstract mathematical entities (platonism)?" a meaningful question, and not an abuse of language?
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Mathematical "forms" as a relation of varying arity

This might be more a MathSE question, but on the other hand, it would involve a peculiar reimagining of the relation between set theory and type theory, so I'll try it out here. OK, so earlier I ...
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Set theory vs. type theory vs. category theory

IIRC, in the univalent-foundations program (per Voevodsky), category theory is represented as a possible sort of evolution or new wave of type theory. Maybe my memory is off, but anyway, in nlab they ...
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107 views

Are there mathematical concepts which we are unable to think of as meaningful representations of real-world things? [closed]

In my limited experience, I cannot think of any mathematical concept which is not obviously linked to the intuitions we have about the real world (irrespective of whether these are actually true or ...
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254 views

How do we arrive at stronger theories in mathematics/logic?

A reasonable aim of formal mathematics/logic is to build systems which can "interpret" many things. As an example, ZFC can interpret a number of things. Incompleteness Theorems provide us ...
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How do logicians think of strength of proof systems?

I want to understand how logicians reason about strengths of proof systems and argue relative strengths of proof systems. I want to appreciate the validity of the reasoning by which we establish ...
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663 views

Do Godel's incompleteness theorems create a contradiction/paradox?

I have seen Godel's theorems presented as a paradox. However, I was only able to infer it's supposed to be one because it proves mathematics to be incapable to be consistent AND complete at the same ...
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133 views

Working of Mathematical Induction

I am aware of what proof by Mathematical Induction is. I have also used it in numerous proofs. However, I don't understand formal correctness/validity of the method down to the level of Peano Axioms. ...
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5answers
184 views

Is there any conflict with Holism and equals and plus signs of mathematics?

Edit - better phrasing/summary: Maybe this phrasing helps "the same object expressed in different ways". That's one meaning behind 'equals'. 10 = 1+...4 --> 10 really is 1+...4. So if ...
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Is defining the concept of Probability still an open problem in the Philosophy of Science?

There exist several interpretations of the concept of Probability: https://en.wikipedia.org/wiki/Probability_interpretations Being the assumption of Repeatability an important difference between them. ...
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Can Mathematics be a tool to analyse immaterial existences

Doubt: Can we say that Mathematical thoughts (arguments) can be independent of physical (Time-space continuum or material) world since it is an abstract science. In other words can Mathematics be a ...
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If most numbers are uncomputable, in what sense do they exist?

Since the set of computer programs is countable and the set of real numbers is uncountable, then it means most real numbers are incomputable. i.e. there does not exist an algorithm to compute their ...
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Technique that "de-trivialising" contradiction (systems)?

Gödel proved that some systems cannot prove their own consistency. As I see, what Gödel proved is no other than that mathematics is freedom, the adventure of a free mind (I.e. not afraid of being ...
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Is there a weaker/general version of Incompleteness Theorem which holds for every formal axiomatic system?

Is there a general version of Godel's Incompleteness Theorem which holds for any formal axiomatic system (and not just those capable of modelling basic arithmetic)? If no, is it absurd to ask why such ...
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Inductive argument for Con(ZFC)

If you ask a mathematician, particularly a set theorist, about whether ZFC is consistent, they will answer that we can't know for sure because of Gödel's theorems. If you ask what evidence at all is ...
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Why can anything be discovered in mathematics at all?

Imagine a Perfect Mathematician that has superhuman abilities -- if you give him or her a formal foundational system for mathematics like ZFC with all the underlying logical machinery, he or she is ...
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Why is the definition of the real numbers not contradictory? [closed]

I understand that a set whose members can, in principle, be enumerated (by having a formula) can be considered as a well-defined set. Therefore, set of all even numbers, multiples of 3, and so on ...
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81 views

Is Constructivism (philosophy of mathematics) against classical logic?

Is Constructivism (philosophy of mathematics) against classical logic? I might be wrong, but mathematics' main branch of logic is based on classical logic, and I was wondering if Constructivism was ...
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What is the current status of Foundation-of-Mathematics programmes?

I have been reading 'A Very Short Introduction to Mathematics' by Timothy Gowers and at one point he mentions that most of the mathematical proofs can be finally resolved to a set of logical ...
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55 views

Is there some non-classical logic where the van der Waerden theorem does not apply?

The van der Waerden theorem is a theorem in the branch of mathematics called Ramsey theory which states that for any given positive integers r and k, there is some number N such that if the integers {...
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What is the connection between Lawvere and Cantor?

Lawvere wrote in a couple papers that Cantors word “menge” which is usually understood as “set” is actually a cohesive type. And the “kardinale” is the abstraction from this by getting rid of the ...
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Symbolic Processes & Thinking

My question is if there is some concrete symbolic logic at the foundation of human reasoning -something very rudimentary, but still formal? Question may be seen in context of the article given below. ...
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Books on Philosophy of Mathematics [duplicate]

I want to buy a philosophy of mathematics book. I have three options in mind: Philosophy of Mathematics by Øystein Linnebo, Philosophy of Mathematics: Selected Readings by Paul Benacerraf, or The ...
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How would a monistic approach account for these categories of probabilities?

Donald Gillies, in his book "Philosophical Theories of Probability," draws a distinction between monistic views and dualistic views of probability, the latter of which, at least in his ...
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What separates mathematics from logic? Can "mathematical" operations be applied to logical systems?

In my 'Introduction to Logic' class, my professor told us that half of the class will be based on "mathematical" operations withing logic. After looking through the textbook, I realized that ...
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Is it possible to create an axiomatic system where 1+1 doesn't equal 2? What would be the consequences of such a system? [closed]

1+1=2 is a result (perhaps arguably more of a definition than a theorem?) of Peano Arithmetic, as well as other systems such as ZFC. I understand that 1+1 doesn't necessarily have to equal 2 if we ...
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363 views

What are the limitations of the language of mathematics?

I was told that mathematics cannot express qualitatively what the elements of a set are, such that you cannot say for example that the members of a set consists of white tigers. So mathematics cannot ...
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58 views

Epistemological Basis of Mathematics Debate

In the following link: https://plato.stanford.edu/entries/intuitionism/ in the last paragraph in Section 1, there is mention of the "lack of epistemological and ontological basis for Mathematics.&...
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Does a Cycle Based Alternative to Set Theory Exist?

My (limited) understanding of Mathematics in general and of Set Theory (being a widely accepted foundational system of Mathematics) in particular, is that the mathematical objects described therein ...
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Non-consistent mathematical axioms

It is known that axioms are the building blocks of mathematics. Differents sets of axioms different "games". What I don't understand is how do we know that we pick axioms that are consistent? . Does ...
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Is there an infinity of axioms in mathematics?

As I was trying to find a list of mathematical axioms used in modern branches of mathematics, I wondered if there's any meaning to the question of "how many mathematical axioms are there ?", and then ...
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450 views

Which problems do you consider as most important open problems in philosophy of mathematics?

At the "intersection" of mathematics and philosophy, or, rather, within their "union", surely some problems are still open and no general consensus is attained when those problems are discussed. ...
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101 views

Is the attempt to separate between Philosophy and Mathematics may be considered as some kind of Philosophy?

When deal with fundamental notions, many mathematicians and some philosophers agree that Philosophy is not an appropriate framework for mathematical frameworks' developments. Is the attempt to ...
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199 views

In non-platonism, can undecidable statements have truth value?

Most sources I can find about Gödel's incompleteness theorems summarize the result as "there exist true arithmetical statements that have no proof." It seems coherent to say that there exist ...
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1answer
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Did Descartes believe arguments for Euclid's parallel postulate were cogent?

If Descartes wanted to found philosophy on the certainty of mathematics, it seems he must have considered arguments for Euclid's parallel postulate cogent, or at least not doubted them. Gerolamo ...
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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 ...
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Why is it argued that an argument has one and only one conclusion?

Why can't an argument have more than just one conclusion? If we assume some premises and we assume them to be true, then by some inference rules we are sometimes able to deduce more than just one true ...
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Why do we need geometry for pure math?

Karl Weierstrass had a very interesting critique of Riemann's work. Supporters of Riemann, claim that a pure logician would never have been able to see the things that the "geometric imagination" of ...
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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 ...
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483 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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 ...
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Do picture proofs of the Pythagorean theorem make it empirical?

As I understand it, the Pythagorean Theorem, which defines the metric for Euclidean space, is said to be strictly mathematical in the sense that it is derived from a set of purely theoretical axioms (...