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The following questions may seem very philosophical and I guess that you guys will tell me that this is not the right place for asking them. But for me it is important to get answers from mathematicians and not from philosophers.

Note: The questions are subjective. I am asking for YOUR opinion. So if you tell me your opinion I would be very happy ;-D

  1. Is "5 is not a set" a mathematical statement? I don't want to know, if you think it is a true statement, or a false statement, but I want to know whether you think it is a statement at all. In type theory, one can't express "5 is not a set". However, should a foundation for mathematics be able to express this sentence?

  2. Is "second-order logic" a logic at all? Or is it just "set theory in disguise"?

  3. Do we use the language of higher-order logic in ordinary mathematics? Or are we always working with first-order logic? Comment: Maybe you are going to say that the definition of a topological space is higher-order because we quantify over sets of sets of ... individuals. But that is not what I mean with "using higher-order logic". The example "definition of topological space" can be expressed in set theory which is a first-order theory. Thus with "using higher-order logic" I do not mean quantification over sets, which is a first-order concept, but I mean "quantification over properties".

  4. Is "working in formal systems (like the predicate calculus)" mathematics? That is: Does "proving of logical tautologies" belong to mathematics? Is it a subfield of mathematics? Is there research in this area?

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closed as primarily opinion-based by Peter Smith, Will Jagy, Dirk, Harish Chandra Rajpoot, Eric Wofsey Sep 22 '15 at 1:12

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Subjective questions tend to be frowned upon here, unfortunately. $\endgroup$ – Omnomnomnom Sep 21 '15 at 16:07
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    $\begingroup$ I know. But where else can I find out the opinions of MATHEMATICIANS to this philosophical questions? $\endgroup$ – sdalfkjasdkölfj Sep 21 '15 at 16:10
  • $\begingroup$ Perhaps we could get away with rephrasing the question as "what are some common opinions on the underlying philosophy" or something like that. $\endgroup$ – Omnomnomnom Sep 21 '15 at 16:11
  • $\begingroup$ Opinion: Yes, it is mathematics, though by no means all of mathematics. $\endgroup$ – André Nicolas Sep 21 '15 at 16:12
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    $\begingroup$ Try asking one question at a time, and things might go better .... $\endgroup$ – Peter Smith Sep 21 '15 at 16:35
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For 1. : Is "5 is not a set" a mathematical statement?

Yes, of course; it is expressible in set theory (like $\mathsf{ZF}$) and set theory, much more than a foundational theory, is a mathematical theory.

For 2. : Is "working in formal systems (like the predicate calculus)" mathematics?

The "mathematical part" of mathematical logic is mainly the meta-theory. In terms of mathematical knowledge, a formal derivation in predicate calculus is quite uninteresting (nor more interesting that performing a sum). But Gödel Completeness Theorem for predicate calculus is an outstanding "piece" of mathematics.

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