On 4/9/2018 7:10 AM, Bruno Marchal wrote:
On 8 Apr 2018, at 19:42, [email protected]
<mailto:[email protected]> wrote:
http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf
<http://iridia.ulb.ac.be/%7Emarchal/publications/SANE2004MARCHAL.pdf>
On 3,) Arithmetic Realism (AR), why is the statement "1+1=2",
equivalent to the Goldbach conjecture, or the inexistence of a
bigger prime, or the statement that some digital machine will stop?
Please take each item in list separately? Goldbach conjecture?
Inexistence? Why stopping? And why can't the statement "1+1=2", just
mean the symbols on the left should be taken to mean the symbol on
the right? TIA, AG
I did not claim that those statement are equivalent, I state only that
they are true of false independently of me.
Then when and if proved they will be automatically equivalent in the
weak sense of classical mathematical logic, where all (known) truth
are equivalent.
The arithmetical realism is just the idea that a (closed) proposition
is either false or truth in the standard model (N, 0, +, *).
But that depends on what "true" means. Whether it refers to fact, a
theorem, or a convention of language.
We can limit realism to the sigma_1 sentences, which can be shown
equivalent with the statements saying that a digital machine stops or
does not stops in arithmetic.
99,9 % of the mathematicians are mathematical realist, which means
that they believe that the excluded middle principle is valid in very
large part of math, like set theory, analysis, etc.
But valid =/= true. It means it prevserves a presumption of true in the
premises...but not that it is the only possible inference rule that does so.
Arithmetical realism is doubted only by ultra-finitist, who believe
that there is no infinities at all, not even at the meta-level.
No. It is doubted by many mathematicians. My mathematician friend Norm
Levitt used to say, "Mathematicians are Platonists Monday thru Friday
and Nominalists on the weekends."
Brent
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