Questions tagged [foundations-of-mathematics]

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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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1answer
395 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 ...
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5answers
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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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4answers
292 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 ...
4
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1answer
100 views

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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3answers
232 views

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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2answers
97 views

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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4answers
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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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3answers
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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 (...
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2answers
126 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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0answers
143 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,...
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4answers
160 views

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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1answer
54 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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3answers
479 views

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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0answers
46 views

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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4answers
314 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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2answers
221 views

Is Mathematical Platonism a meaningful thing? [closed]

How can we be sure that asking If Mathematics is Platonic or If it is Our Own Construction is a meaningful thing to ask, and not just abuse of natural language? Can somebody provide a convincing ...
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3answers
141 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 ...
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1answer
149 views

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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0answers
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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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82 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 ...
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0answers
85 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 ...
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2answers
217 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 ...
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88 views

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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2answers
162 views

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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1answer
264 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 ...
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65 views

Why is Entscheidungsproblem problem undecidable? [duplicate]

I have been trying to wrap my head around Gödel's Completeness Theorem, Incompleteness Theorem(s) and undecidability of Entscheidungsproblem. Turing wrote: It should perhaps be remarked what I shall ...
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120 views

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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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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331 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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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 ...