28
votes
Accepted
Why do universities not teach constructive mathematics to CS undergraduates?
Let me offer a few thoughts, specific to mathematical pedagogy in computer science (in particular for the states):
(a): a typical BS computer science program barely has time to touch on computational ...
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16
votes
Why do universities not teach constructive mathematics to CS undergraduates?
So we're clear, mathematical constructivism is a logic/philosophic approach to conceiving mathematical activity. That's pretty remote from undergraduate CS pedagogy. From WP:
In the philosophy of ...
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16
votes
Why do universities not teach constructive mathematics to CS undergraduates?
"Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “we can construct”."
The ...
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14
votes
Accepted
In simple terms, what is the difference between logic in mathematics and philosophy?
The definitions of 'logic' and 'mathematics' are themselves subject to dispute. In particular, the word 'logic' is used in different senses. At its narrowest, it is concerned with the relationship of ...
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12
votes
Accepted
Is mathematical creativity the same as artistic creativity?
There are caricatures of math and the arts, and then there are characterizations, the best of which are accurate. Many students get dragged through the drudgery of mathematical algorithms and washout ...
- 23k
11
votes
Accepted
Set theory vs. type theory vs. category theory
Short Answer
It sounds you're struggling to understand the relationship between three fundamental theories. Naive set theory is the theory used historically by Gottlob Frege to show that all ...
- 23k
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 ...
- 23k
8
votes
Accepted
What is the difference between logic and mathematics?
Whether there is a distinction, and what the distinction consists in, is a hotly debated topic. Here are a few things that are typically claimed to be essential to logic:
Universal applicability: the ...
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7
votes
Accepted
What are the main issues on which the schools of Intuitionism, Formalism, and Logicism disagree?
Here is partial preliminary answer:
Logicism, intuitionism and formalism are three traditional views about the nature of mathematics.
Formalism was introduced by the German mathematician David ...
- 102
6
votes
Do Godel's incompleteness theorems create a contradiction/paradox?
The short answer is that Gödel's incompleteness theorems are not contradictory, and arguably they are not paradoxical either, except in so far as they upset our preconceptions about provability and ...
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6
votes
Why do universities not teach constructive mathematics to CS undergraduates?
Due to many reasons, main ones are follows:
It is computer science program, and not philosophy program. Certainly, foundations are in philosophy (mathematical logic), but curriculum focusses on well-...
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6
votes
Why do universities not teach constructive mathematics to CS undergraduates?
So there's something which might be worth discussing here; namely, given that the primary dispute between constructivists and classical mathematicians relates to the transfinite, does the dispute ...
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6
votes
Accepted
Why can Goedel's Incompleteness Theorem be proven?
"Godel thus proved an unprovable statement" - this is not quite the case, as you yourself recognize in the lines following. Rather, Godel proved that there exists an undecidable statement ...
- 2,099
5
votes
Why do we need geometry for pure math?
Newton, with the bias of his time, sought to formulate all his proofs in geometrical terms. He was also a freemason, a cult-tradition that identifies divinity with architecture, and geometry - and ...
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5
votes
Accepted
What's so bad about giving up the Axiom of Choice?
There is a lot of writing both in favor and against AC from a philosophical standpoint - e.g. in favor see Penelope Maddy's Believing the axioms.
However, there are also more mundane issues. I think ...
- 3,210
5
votes
Is it possible to create an axiomatic system where 1+1 doesn't equal 2? What would be the consequences of such a system?
Your question is grammatically sound but it sort of defeats itself by suggesting to rewrite definitions but then judge them by the old definition. I'm not accusing you of any illegal substance abuse, ...
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5
votes
If most numbers are uncomputable, in what sense do they exist?
Computability of real numbers from Turing machine or Church's lambda calculus isn't necessary for a generic existence. According to computable number reference here:
Every computable number is ...
- 3,843
5
votes
Is mathematical creativity the same as artistic creativity?
Yes, in essence.
For my argument I will consider mathematicians only as creators of mathematical stuff, not just copiers and learners. After all we don't call proof-readers and copy-typists authors.
...
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4
votes
Why can anything be discovered in mathematics at all?
I am not entirely certain what exactly you mean, but here are some thoughts which might qualify as an answer.
The enterprise of mathematics makes use of both discovery of unexpected results and ...
- 7,790
4
votes
Accepted
Set theory with full comprehension
In general, what you are referring to are paraconsistent set theories. These are versions of set theory in which the underlying logic lacks the principle of explosion, and in some cases tolerates true ...
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4
votes
Accepted
Does Münchausen's trilemma apply to mathematics?
Münchausen's trilemma is often presented as a kind of skeptical argument, purporting to show that knowledge, or demonstration, or certain belief, or something related to these, is impossible. But it ...
- 22.9k
4
votes
In simple terms, what is the difference between logic in mathematics and philosophy?
Caveat
The definitions of 'logic' and 'mathematics' are themselves subject to dispute. In particular, the word 'logic' is used in different senses. - Bumble
Bumble's answer is great, and mine is ...
- 23k
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,” ...
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3
votes
Accepted
Did Descartes believe arguments for Euclid's parallel postulate were cogent?
Descartes considered that he needed Euclid's parallel postulate:
Descartes identified space and
the extension of matter, so geometry was, for him, about real physical space. But *geometric ...
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3
votes
Accepted
Inductive argument for Con(ZFC)
I would respond that the question is moot: it is too difficult (at present) to talk in a meaningful way about the probability of a mathematical statement.
This is because one of the standard ...
- 3,210
3
votes
Why is the definition of the real numbers not contradictory?
This is a misconception, you don't need to be able to enumerate the elements of a set.
In naive set theory (which has its problems but is useful to explain the set concept here), a set is defined by ...
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3
votes
Why is the definition of the real numbers not contradictory?
Most mathematicians are happy to use ZFC set theory or one of it's equivalents. These set theories support the "normal" real numbers.
There are, however, mathematicians such as the ...
3
votes
Accepted
Why can anything be discovered in mathematics at all?
None of the other answers (so far) addressed the actual mathematical inquiry here.
Firstly, your question is straightforwardly formalized as follows: Suppose you have an oracle O that when given any ...
- 546
3
votes
Is there any conflict with Holism and equals and plus signs of mathematics?
Long comment
When in mathematics we write 1+2+3+4=10 we are not making some sort of "metaphysical claim": we are asserting that when we evaluate the left-hand side expression (we "...
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Only top scored, non community-wiki answers of a minimum length are eligible