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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 ...
emesupap's user avatar
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17 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 ...
Jo Wehler's user avatar
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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 ...
J D's user avatar
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13 votes
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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 ...
J D's user avatar
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13 votes
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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 ...
Bumble's user avatar
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12 votes
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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 ...
J D's user avatar
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12 votes

What is a natural number?

As far as I can tell, Peano's Axioms arose from properties of the counting (natural) numbers that have been known for thousands of years: Zero is a natural number Every natural number has a unique ...
Dan Christensen's user avatar
10 votes

What is a natural number?

Begin with the idea of being. The being can be of an abstract mathematical object like Boolean TRUE, a finite instruction like "step forward", or an identifiable physical process like an ...
g s's user avatar
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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 ...
J D's user avatar
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8 votes
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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 ...
Dennis's user avatar
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8 votes

What is a natural number?

This is an old and subtle question in philosophy of mathematics, but as a starting point, it’s hard to do better than the classic Benacerraf 1965 paper, What numbers could not be. It’s very readable —...
Peter LeFanu Lumsdaine's user avatar
7 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 ...
Bumble's user avatar
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7 votes
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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 ...
emesupap's user avatar
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6 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 ...
Double Knot's user avatar
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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-...
Ajax's user avatar
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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 ...
Paul Ross's user avatar
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6 votes

What is a natural number?

Set theory such as ZF is a customary framework in which one expects all higher-level objects to be developed, but the particular construction such as von Neumann's shouldn't be taken as "the ...
Mikhail Katz's user avatar
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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, ...
Flater's user avatar
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5 votes
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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 ...
Noah Schweber's user avatar
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 ...
CriglCragl's user avatar
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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 ...
niels nielsen's user avatar
4 votes
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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 ...
Bumble's user avatar
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4 votes
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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 ...
Bumble's user avatar
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4 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. ...
Atif's user avatar
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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,” ...
Jo Wehler's user avatar
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3 votes
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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 ...
Geoffrey Thomas's user avatar
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3 votes
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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 ...
Noah Schweber's user avatar
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 ...
kutschkem's user avatar
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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 ...
Math Keeps Me Busy's user avatar

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