9
votes
Is mathematics analytic or synthetic?
A possible counterargument is that the analytic-synthetic distinction you are using is inherently inadequate and outmoded language and thinking. For the first part, Quine in his Two Dogmas of ...
- 22.9k
8
votes
If Large Language Models can do Maths, is Formalism true?
As a constructivist brother who places as much credence in Platonic Forms as he does in the Irish tuatha da dannan or the Norwegian troll, let me dispute the premise that LLMs do math or have much in ...
- 22.9k
5
votes
Are laws separate “objects” or are they inextricably part of the universe?
You say :
Suppose that the universe is deterministic because of some laws. But
those laws themselves exist for no reason.
Here, I see three assumptions :
The universe is deterministic.
The ...
- 1,680
4
votes
Accepted
Is Fermat's last theorem a logical necessity or a different kind of necessary truth?
Strictly speaking it can be proved using a combination of the mainstay mathematical axioms, logical rules, and syntax. Mathematics is not known to be “reducible” to logic, so this can’t be a logical ...
- 2,385
4
votes
Is Fermat's last theorem a logical necessity or a different kind of necessary truth?
The proof of Fermat’s last theorem, i.e. of the theorem of Wiles, was very challenging. But concerning its state as a mathematical theorem, Wiles’ theorem does not differ from all other mathematical ...
- 24.4k
4
votes
Is mathematics analytic or synthetic?
The two terms, analytic and synthetic, are two possible, mutual exclusive
properties of statements. SEP introduces the following definition:
“Analytic” sentences, such as “Pediatricians are doctors,” ...
- 24.4k
4
votes
If Large Language Models can do Maths, is Formalism true?
Dougherty[95] is a continuation of an examination of a topic in the theory of large cardinals (those which are critical points of elementary embeddings) that is some many years old, the abstract for ...
- 12.1k
4
votes
If Large Language Models can do Maths, is Formalism true?
IMO we may consider the link between math and language (maybe more... maybe math is language).
Consider a ChatGPT answering our questions. What is it doing? Is it speaking? Or it is only simulating a ...
- 36.2k
3
votes
Is mathematics analytic or synthetic?
There are various ways to define "analytic" and "synthetic". Those word are generally thought to apply to propositions, but there are different ideas of what a proposition is. ...
- 7,600
3
votes
Is mathematics analytic or synthetic?
For the sake of the OP, I will assume that some version of the analytic/synthetic distinction is defensible. More specifically, I will assume that we can differentiate between analyzing a question ...
- 12.1k
3
votes
Are laws separate “objects” or are they inextricably part of the universe?
Any time you are discussing the "ultimate nature of things", you're firmly in the domain of metaphysics.
The answer you seek depends on your metaphysical presuppositions. Many are content to ...
- 22.9k
2
votes
Are laws separate “objects” or are they inextricably part of the universe?
You ask the old question: Do we discover scientific laws, or do we invent scientific rules to explain our observations?
The question is old. Until now it has no answer.
Lessons learned: What in the ...
- 24.4k
2
votes
Is Fermat's last theorem a logical necessity or a different kind of necessary truth?
Mathematics has a set of axioms (statements that we call "true", and that cannot be constructed from other axioms), and a set of rules that allow us to construct more statements from these ...
- 5,243
2
votes
What is it that is done when we DO mathematics?
What are the deeper contents of the doing of mathematics?
Mathematicians invent concepts from the domains below, state some
interesting axioms and prove by applying logical rules, that certain
...
- 24.4k
1
vote
What is it that is done when we DO mathematics?
As a first stab at an answer, based on the operations of my own mind, I would argue that if the mind is not recognizing the sense of proportion and/or the concept of numbers then it is not doing math. ...
- 644
1
vote
What is it that is done when we DO mathematics?
One answer is provided by Leibniz in terms of his idea of mathematical entities as "useful fictions". The view of mathematical entities as fictional was propounded by mature Leibniz no ...
- 906
1
vote
Accepted
What is it that is done when we DO mathematics?
Short Version
To me the central thing about math is that it is
platonism-reified.
When a mathematician does mathematics what they are doing is performing that reification.
Note the fine dance that's ...
- 1,359
1
vote
What does philosophy have to do with category theory?
A good point-of-entry here is the SEP article on category theory, specifically the section on its philosophical significance:
Category theory challenges philosophers in two ways, which are not ...
- 12.1k
1
vote
Are laws separate “objects” or are they inextricably part of the universe?
The Universe behaves in a characteristic way, in which we perceive ubiquitous patterns. The laws of physics are labels and symbolic representations for the more fundamental of those patterns. For ...
- 13.5k
1
vote
Are laws separate “objects” or are they inextricably part of the universe?
While I am not an expert in the field of mathematics and/or gravitational physics I can tell you one thing. You mustn't forget that laws themselves are objects of some sort the second you think of ...
- 11
1
vote
Is Fermat's last theorem a logical necessity or a different kind of necessary truth?
Excellent question, and one in which there may not be a mutual exclusion.
Obviously, if a theorem is true, it is logically necessary in respect to the axioms of the system. That's the easy half of ...
- 22.9k
1
vote
Are Bourbaki and Deligne Mathematical Realists?
Bourbaki have insisted that they are interested in the way
mathematicians do their work rather than in foundations, and there are
indications that their philosophy of mathematics is not carefully
...
- 906
1
vote
Are Bourbaki and Deligne Mathematical Realists?
Concerning your first question see Nicolaus Bourbaki The
architecture of mathematics. The paper is a self-presentation of Bourbaki from 1950, on request I can send a copy.
Reading the paper confirms @...
- 24.4k
1
vote
Omniscience leads to necessitarianism
Let's work in a temporal logic with five tenses Pa, Pr, F, N, and Æ: "It was true that," "It is true that," "It will be true that," "It is never true that," and,...
- 12.1k
1
vote
Accepted
Gödel's Asymmetry
Since axioms, when true, are true apart from proof, there must be some discrepancy within theories of proof and between provability and truth. If Tarski had not put the liar paradox to use in a way ...
- 12.1k
1
vote
Looking for a reference on a kind of mathematical platonism
James R. Brown is a philosopher at the University of Toronto who tends towards Platonism. His works are easy to read; you might want to check out his books "Philosophy of Mathematics: An ...
1
vote
Why do we have a problem about understanding the concept of the "empty set"?
There are two separate questions here: why do we have difficulty with the empty set; and why do we need the existence axiom. The well-known principle of mathematical induction includes two clauses: (...
- 906
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