Questions tagged [foundations-of-mathematics]

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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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145 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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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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2answers
201 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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0answers
91 views

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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50 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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84 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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90 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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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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45 views

Why did Voevodsky feel “objects of a category can never be equal”?

In a lecture that Voevodsky gave at the IAS on the different notions of equality. He specified how this came from people who worked with categories and their higher analogues. The main problem he ...
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54 views

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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124 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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1answer
431 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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1answer
90 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 ...