# Re: The Origin of Physical Laws and Sensations - Bruno Marchal, Université de Bruxelles

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On 4/9/2018 7:10 AM, Bruno Marchal wrote:
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On 8 Apr 2018, at 19:42, agrayson2...@gmail.com <mailto:agrayson2...@gmail.com> wrote:
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http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.pdf <http://iridia.ulb.ac.be/%7Emarchal/publications/SANE2004MARCHAL.pdf>
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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
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I did not claim that those statement are equivalent, I state only that they are true of false independently of me.
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Then when and if  proved they will be automatically equivalent in the weak sense of classical mathematical logic, where all (known) truth are equivalent.
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The arithmetical realism is just the idea that a (closed) proposition is either false or truth in the standard model (N, 0, +, *).
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But that depends on what "true" means.   Whether it refers to fact, a theorem, or a convention of language.
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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.
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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.
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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.
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Arithmetical realism is doubted only by ultra-finitist, who believe that there is no infinities at all, not even at the meta-level.
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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."
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Brent

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