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13
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4answers
1k views

What are computable numbers, and what is their philosophical significance?

What are Computable Numbers? Is computability (or non-computability) some sort of technology-dependent characteristic of numbers (via e.g. Turing Machines)? What are the philosophical implications or ...
9
votes
3answers
215 views

Is there a way to avoid Gödel incompleteness affecting mathematics as a whole?

I have been thinking about Gödel's incompleteness theorems and their ramifications for the whole of mathematics. In this question I assume some fixed formal system F expressive enough for the ...
2
votes
3answers
61 views

What is(are) the importance(s) of formal reasoning

In Mathematics, we as an undergraduate are exposed for the first time (at least for me it was the case) to 'rigor'. For example, in Real Analysis classes we often use logical quantifiers in our ...
4
votes
2answers
210 views

What are the “undefinable numbers” in real analysis and philosophy?

What if any important results in real analysis make use of the notion of an "undefinable" real number? (Whatever "important" may mean to the reader.) Or is it used more in the philosophy of ...
1
vote
4answers
79 views

Does Philosophy have a Good Answer to Personal Inadequacy?

Some of the answers and comments to this question got me thinking about the problem of personal inadequacy. By this I mean, people seem to regularly hold values/ideals/morals beyond anything they ...
8
votes
1answer
143 views

Where did Gödel write that first-order logic is the “true” logic?

In "On How Logic Became First-Order" Matti Eklund writes (p. 2/148): It appears to be widely held today that arguments from Skolem and Kurt Gödel, both alleged proponents of the thesis that ...
8
votes
1answer
197 views

Was there a Kantian influence on Hilbert's formalist programme?

In this paper by Cassou-Nogues which is on an aspect of the mathematical philosophy of Cavailles he quotes the mathematician Hilbert (a colloborator of Einstein in Gottingen) ...We find ...
2
votes
2answers
121 views

Prove a logical formula is equivalent to the contradiction if and only if the set it describes is empty

Let ψ be a well-formed-formula (wff). Prove that (ψ ≡ ⊥) ⇔ {x:ψ(x)}=Ø that is, the formula ψ is a contradiction if and only if the set it describes has no members. Note This question is not about ...
5
votes
1answer
222 views

Are there rules for dealing with self-reference “paradoxes” in logic?

My favorite paradox that leads to an endless regress, and also leads to a question: The sentence after this is true. The sentence before this is false When contradictions appear in proofs, ...
3
votes
3answers
255 views

What is the minimum number of axioms required for a system of axioms?

What is the minimum number of axioms you need, apart from definitions and usage of the notation, such that you have a system that does not contradict itself? I would just think that the answer is ...
2
votes
1answer
63 views

Transitive Incompleteness of Logical WFF's Due to Godel's Incompleteness Theorem

If a set of theorems, or wff's, are used in conjunction with one another, does this have an impact on their completeness in terms of soundness? For example, I have five theorems of logic, or ...
4
votes
2answers
306 views

How does abstraction/generalization in mathematics fit into inductive reasoning?

I have a question about the nature of generalization and abstraction. Human reasoning is commonly split up into two categories: deductive and inductive reasoning. Are all instances of generalization ...
3
votes
2answers
219 views

Definition of concept

What definition does contemporary analytic philosophy give of 'concept'? And what is the difference between a concept of something and a conception of something? Then what's the difference between ...
1
vote
0answers
56 views

What are the biggest advances in formal philosophy? [closed]

e.g. Is the formalization of occam's razor as minimum description length (Solomonoff induction) an example of this? Or the formalization of complexity as kolmogorov complexity?
3
votes
0answers
96 views

Logic as a (bad) model language

Can someone name the most well known philosophers to explicitly put forward an idea along the lines that formal systems can only be used descriptively, not prescriptively - that they're just a model, ...
2
votes
1answer
129 views

What are the ramifications of the limitations of ZFC set theory?

In the Wikipedia article on Zermelo-Fraenkel set theory says that the theory sets out to formalize a notion of sets such that "all entities in the universe of discourse are such sets." It goes on to ...
7
votes
2answers
256 views

How does “higher-order logic” differ from “normal” (first order?) predicate logic?

How does “higher-order logic” differ from “normal” predicate logic? I assume the latter is consistently called ”first order logic”. So where are the differences between these? What kinds of statements ...
26
votes
2answers
1k views

Is First Order Logic (FOL) the only fundamental logic?

I'm far from being an expert in the field of mathematical logic, but I've been reading about the academic work invested in the foundations of mathematics, both in a historical and objetive sense; and ...
6
votes
2answers
198 views

Is there a formal presentation of Maturana's “Autopoiesis”

I have been reading through "Autopoiesis and Cognition" by Humberto R. Maturana and Francisco J. Varela. One of their goals in defining autopoiesis and the supporting concepts of simple and composite ...
18
votes
11answers
1k views

Is it possible to scientifically determine good and evil?

Sam Harris has argued on many occasions - the earliest of which I'm aware of being in his book, The End of Faith, as well as later on in The Moral Landscape - that it is (at least theoretically) ...
21
votes
4answers
6k views

What are the philosophical implications of Gödel's First Incompleteness Theorem?

Gödel's First Incompleteness Theorem states Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any ...