# Questions tagged [foundations-of-mathematics]

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### How do we determine if a statement can't be proved by mathematics alone?

How do we determine if a statement can't be proved by mathematics alone? It seems mathematics can only prove something that can be defined purely in mathematics terms, but can't prove simple ...
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### Why can't mathematics prove that Tic-tac-toe is a simple game? [closed]

Why can't mathematics prove that Tic-tac-toe is a simple game? Tell me if I am wrong, but we can prove that chess is less complex than tic-tac-toe, by counting the number of actions possible if we ...
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### Why can Goedel's Incompleteness Theorem be proven?

Preliminaries: The way I understand it, Gödel took Russel's and Whitehead's Principia Mathematica (PM) and mapped strings of symbols from PM onto the integers, their Gödel numbers. He then constructed,...
171 views

### Why do people still use classical logic? [closed]

It seems to me very crazy that mathematicians reactions to Godël incompleteness theorem have been mostly to agree that there are statements of the language which can neither be proved nor disproved ...
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### I am looking for some ternary logic with values True/False/Absurd

I am looking for some alternative logic where a sentence p could not only be true or false but also absurd, which is different of false in such a logic. I see that there is some content about three ...
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9k views

### Why do universities not teach constructive mathematics to CS undergraduates?

I had a conversation with a user on the Internet. And it did indeed wake my interest regarding something that I had also been asking myself long ago. Why do so many universities still teach beginners ...
141 views

### Per Mathematical Structuralism, can a pure mathematical theory have semantics that is not closed on isomorphism?

This question is the philosophical side of a question that I've recently posted to MathOverflow. Here, I'm specifically asking about the output of Mathematical Structuralism on that question that I'll ...
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### Do any philosophers say surreal numbers are reason to doubt platonism?

Not trying to be inflammatory at all, this is a genuine (maybe dumb) question. Especially in regards to the genesis of the surreals, which was Conway thinking about Go endgames. They seem among the ...
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38 views

### Realism as necessary for impredicative mathematics to avoid viscous circle, but not really?

Here is an quote from Godel from Shapiro’s Thinking About Mathematics: “…the vicious circle…applies only if the entities are constructed by ourselves. In this case, there must clearly exist a ...
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167 views

### What is mathematical analysis?

Hilbert's aim to reduce all mathematics to finite logical system was shown impossible by Goedel. He did mathematical analysis of logic itself (Goedel numbering). Turing defined algorithms, and ...
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1 vote
136 views

### Two questions about mathematical platonism

Any set, number, shape, definition, axiom, etc we write down or think about is not the ideal platonic version. But surely the mathematical platonist thinks humans are closer to that unreachable goal ...
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1 vote
72 views

### Is there a naturalized intuitionist mathematics? Is it Kantian?

I have in mind an interpretation of mathematics as intuitionalism, where intuitions are subjective (built from personal experience), but subjective experience is ultimately explained “objectively” a ...
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### In simple terms, what is the difference between logic in mathematics and philosophy?

I want to understand the difference between mathematical and philosophical logic. I actually thought they were the same till I read this post. Concisely speaking, what is the difference between how a ...
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### Does Münchausen's trilemma apply to mathematics?

I'm a mathematician/statistician, and I've been recently reading about epistemology and philosophy of science in my particular field of study. In statistics, there is a deep concern for the objective ...
506 views

### Set theory with full comprehension

A few years ago I was reading an entry in the Stanford Encyclopedia of Philosophy related to set theory and I stumbled across a statement along these lines: There is a set theory where full ...
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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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94 views

### 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 ... 46 views

### 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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1 vote
93 views

### 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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681 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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### 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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201 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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90 views

### 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 ...
320 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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473 views

### 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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77 views

### 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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### 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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119 views

### 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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100 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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1 vote
141 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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1 vote
61 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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134 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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### 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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### 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 ...
729 views

### 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 ...
226 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 ...
406 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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### 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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### 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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### 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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### 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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### 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. ... 112 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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### 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 ...
253 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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### 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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