31
votes
Is there just one Zero?
In group theory, the "zero" is better referred to as the identity element, usually within what are additive groups.
This is not the same in every group.
For matrices, it is the n × n zero-...
- 5,460
15
votes
How do mathematical realists explain the applicability and effectiveness of mathematics in physics?
Physics doesn't follow any rules. Physics just is. If physics stops following our rules, we don't say physics is wrong. We have to adapt our rules.
Many of the rules and patterns we discover in ...
- 2,243
12
votes
How do mathematical realists explain the applicability and effectiveness of mathematics in physics?
why does the physical universe adhere to mathematical principles?
It might not. The adherence is generally understood to be the other way around. We have proposed mathematical models that adhere to ...
- 5,460
10
votes
Is there just one Zero?
Try holding only a book in your hand, and asking them how many pens you're holding.
Zero doesn't exist at all. Neither does one, two, three, etc.
These are just concepts we often (but not necessarily)...
- 13.6k
8
votes
How prevalent is the formalist perspective on mathematics?
See Philip Davis and Reuben Hersh's quote [page 359 of the 1995 Edition of The Mathematical Experience (1981)]:
“The typical working mathematician is a Platonist on weekdays and a Formalist on ...
- 41.1k
8
votes
How prevalent is the formalist perspective on mathematics?
I imagine many mathematicians haven't given the matter much thought at all, but among those who have, the great divide is in the attitude toward an independent realm, or independent reality, of ...
- 4,015
7
votes
Is there just one Zero?
There are many related issues:
the symbol 0: different in different languages/notations,
collections of objects with the same number of objects: how collections of NO objects at all can differ? and ...
- 41.1k
7
votes
Are mental images of mathematical entities persistent?
The view that mathematical objects are mental models is called psychologism. It is not a popular view of mathematics for several reasons. Perhaps the most pressing is that it implies that ...
- 13.1k
7
votes
Has Giaquinto been refuted?
I think Giaquinto makes a subtle fallacy by assuming incompleteness is a mark against formalism exclusively. In a certain sense, incompleteness is really just the halting problem in disguise. The ...
- 249
6
votes
How do we justify the Power Set Axiom?
The motivation, according to the iterative conception of sets - see also Von Neumann universe - is that sets "come into being" at stages.
There is an initial collection of individuals: ...
- 41.1k
6
votes
Is there just one Zero?
I am reminded of Wittgenstein, because ambiguities like these where all the words seem different but seem to reflect no underlying difference in reality are really his jam in Philosophical ...
- 161
5
votes
Is there just one Zero?
There are three zeros in 1000, but they mean different things: zero units, zero tens and zero hundreds.
- 2,243
5
votes
Are mental images of mathematical entities persistent?
Two books that might be relevant:
Mathematical Epistemology and Psychology, by E.W.Beth and Jean Piaget
This is a remarkable book - product of the cooperation between a Dutch mathematician/logician/...
- 4,784
5
votes
How do we justify the Power Set Axiom?
Welcome to the brave new world of Constructivism (Errett Bishop) and Predicativism (Hermann Weyl, Solomon Feferman, ...). These schools indeed reject the generalized application of the power set ...
- 4,015
4
votes
How do we justify the Power Set Axiom?
The Power Set Axiom is the set theoretic Analogue To Exponentiation.
The same way the Union and Pairing Axioms allows us to have a successor of a set.
The Power Set let's us have exponentiation of ...
- 472
4
votes
Accepted
What is the ontological difference between platonism and non platonism?
Translated to mundane speech Platonism means that certain concepts like mathematical objects exists independently from human thinking. These concepts can be discovered by humans, but they are not ...
- 42.3k
4
votes
Is there just one Zero?
This is a corollary of the fact that there exists exactly one empty set: {}. Therefore any intensional formula that defines a condition that no object satisfies, such as the three examples you gave, {...
- 973
4
votes
Is there just one Zero?
There is exactly one 'zero' in the same sense that there is exactly one 'one' or one 'two'. Numbers are linguistic quantifiers that call for a reference object. The reason people think that 'zero ...
- 24.1k
4
votes
How can we prove that PA has a model?
Double Knot noted that, per your primary question, the issue is relative consistency. We might also call it "conditional" consistency: if S is consistent, then from S we can prove that T is ...
- 22.4k
4
votes
How do mathematical realists explain the applicability and effectiveness of mathematics in physics?
I think that there is an underlying assumption that we "found" mathematics and that it just "happens to" align with physical reality, but there is no reason to believe that this is ...
- 321
3
votes
Are mental images of mathematical entities persistent?
You mentioned last time the essay “Bernays, Paul. Zum Symposium "Ueber die Grundlagen der Mathematik. Dialectica 25, no. 3/4,”
I stumbled on his remark on p. 177:
Zuzugeben ist, dass wir keine
...
- 42.3k
3
votes
Did Bernays change his mind?
See Paul Bernays, Sur le platonisme dans les mathématiques (1935).
From the English translation [all emphasis mine]:
An example of this way of setting up a theory can be found in Hilbert’s ...
- 41.1k
3
votes
What is the ontological difference between platonism and non platonism?
Historically, mathematical Platonists may have had an idea of mathematical entities being present in space, but today (after general relativity, quantum mechanics, etc.) it would be difficult to ...
- 4,015
3
votes
Quantitative definition of information without set cardinality or probability?
You know that the number x in question is a natural number. Considered in the dual system x is represented as a finite sequence of digits 0 and 1.
If you are told the first digit you got 1 bit of ...
- 42.3k
3
votes
What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?
The answer is laid out clearly in Cantor's 1890 letter to Veronese. In an earlier letter, Veronese had quoted Otto Stolz to the effect
that Cantor's results have no bearing on Stolz's infinitesimals, ...
- 4,015
3
votes
How can we prove that PA has a model?
"if Con(ZFC) was false, would the proof of the consistency of PA still be valid?"
Possibly yes, because existing proofs of consistency of PA require far less than Con(ZFC); notably Gentzen'...
- 4,015
2
votes
How prevalent is the formalist perspective on mathematics?
Complementing @Mauros quote: IMO the working mathematician in pure mathematics does not care about Platonism or formalism, not at all and not even on Sundays :-)
If necessary and if he/she is a ...
- 42.3k
2
votes
How do we justify the Power Set Axiom?
Cantor derived the assertion that ℕ and ℝ are not equinumerous prior to his generalized powerset theorem. If you don't have choice, you can bring in amorphous sets, whose cardinalities are uncountable,...
- 22.4k
2
votes
Quantitative definition of information without set cardinality or probability?
The Stanford Encyclopedia of Philosophy article on generalized quantifiers includes a section that opens with the following:
Other properties are not shared by all natural language quantifiers but ...
- 22.4k
2
votes
Is Lowenheim-Skolem just an artifact of the formalism?
The downward version of the theorem could be construed as "just an artifact" of the formalism of first-order logic and its attendant theories, but since there is nonfirst-order logic (e.g. ...
- 22.4k
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