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After investigating dimensionless physical constants, I've been receiving lots of criticism from scientists, specifically physicists, that mathematics is not science. Is there a clear distinction between science and mathematics that could justify a scientist saying that a mathematical idea is not science?

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    The issue is largely terminological. Uncontroversially, mathematics is not an empirical science (it uses proofs in place of experiments), so the first question is whether one uses "science" for "empirical science" only, or also includes formal sciences. Either way, the two have similarities and differences, and the second question is whether to count that as half-full or half-empty, see What makes something mathematics? – Conifold Sep 29 '17 at 23:11
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    Of course, a lot of science is non-empirical, though, and not just formal sciences—some theoretical ones. – ChristopherE Sep 30 '17 at 0:23
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    Mathematics is not a science and exists outside of any physical reality. You could create mathematics entirely within your mind with no knowledge that there is an outside world. In other words, "I think therefore I am" is sufficient to create mathematics. The fact that mathematics can be used to describe the physical world is a nice bonus, but pure mathematics is an entirely symbolic system. – barrycarter Sep 30 '17 at 13:53
  • @ barrycarter: How could a brain empty of all impressions from outside devise anything? How would it be able to imagine a triangle? What do you think when you imagine a triangle? A picture, a body, three points in distances to each other? Same with 2 + 3. Do you imagine an axiom, perhaps from Peano that not even defines integers except for the first? No you imagine physical bodies. Without this application of physics no mathematics would have been created at all. And if yet, it would be as irrelevant as astrology and certainly not taught at universities. – Heinrich Oct 1 '17 at 10:12
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Originally mathematics, namely Euclidean geometry, counting and the four basic arithmetical operations is a branch of physics. The basic activity is finding labels (numbers) for sets of material bodies while properties like shape, mass, colour, etc. are disregarded. Same activity can be observed in other sciences, for instance classifying in botany or geology. All results of this mathematics can be experimentally verified.

Also higher mathematics like analysis belongs to physics and sciences. This understanding has prevailed until far into the 19th century as can be seen from the fact that most universities have faculties of "sciences and mathematics" and mathematicians have lectured about theoretical physics (Cantor for instance lectured about mechanics).

In principle the whole contents of mathematics can be reduced to this scientific basis, namely to the handling of integers - as far as real mathematics is concerned. But mathematics without abbreviations would be very elaborate and tedious.

Simple examples: 2^3 = 2*2*2 = (2 + 2) + (2 + 2), and 2 = { } U {{ }} U {{ } U {{ }}}.

Harder examples 7^7^7 = ..., and 7 = ...

Therefore lots of abbreviations have been invented. And some mathematicans believe that these abbreviations stem from or belong to a "higher sphere". Like priests of the god of thunder have thought about their profession.

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    Just as a sidenote: I heard academic mathematicians state that handling of integers could hardly be called mathematics and true mathematics would start where it gets rid of any particulars. – Philip Klöcking Oct 1 '17 at 10:17
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    I would call these colleagues aufgeblasene Wichtigtuer. – Heinrich Oct 1 '17 at 14:40
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At this time, nobody can say for sure what the physics/math difference is. That is, to what extent the foundational elements of physics can be entirely generated from purely logical/mathematical considerations, and to what (if any) extent there's an irreducible kernel of ad hoc empirical fact that has to be axiomatically introduced.

So, you don't say exactly what your "investigating dimensionless physical constants" consists of. But your question strongly suggests you're trying to establish a mathematical relation between (some of) them that reduces (if not eliminates) the empirical input necessary to describe nature (e.g., gravity related to electromagnetism). That's not a new idea, e.g., Dirac's Large Number Hypothesis, https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis And Dirac's idea hasn't "been receiving lots of criticism", though it's never received lots of active investigation, either. So, describe exactly what you're doing, and perhaps its deserved or not-deserved criticism will be more apparent.

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    Thank you for the response. What exactly I've been doing is observing, IMO with some empirical observation, that the number 1 is a fundamental physical constant. Where f(r) = (ccr/Gm) and 'A' = mtr, the limit of f(r) as A approaches 1 (imagine the simplest electromagnetic vacuum) exists at 1. f(r) reduces to 1,346,634,684,151,322,059,505,072,689.31... The constant was derived from the 2014 CODATA values. The number is derived from dividing e=mc^2 and F=Gmm/r^2 because of the relationship in Fs=W. Even if the physics is questionable, I'm asking if it's possible for "1" to be a sci/math bridge. – user149553 Sep 30 '17 at 12:46
  • Sorry, looks pretty criticizable to me. What's t,s,W supposed to be? And what m are you choosing (since your e/F ratio leaves an extra m unaccounted for)? Moreover, that ratio seems unrelated to your f(r). And what's the significance of your ~1.3x10^27 (which you give with more significant digits than any current physical measurement)? So what if f(r)-->1.3x10^27 as A-->1??? That 1.3x10^27 is a meaningless number, as far as I know, whereby the entire exercise seems meaningless to me. And that's even if you'd defined all your terms properly, which you haven't, as far as I can tell. – John Forkosh Oct 1 '17 at 1:02
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    t = time, s = distance(in this case, a radius), W = work(in this case energy), the constant G has has the dimensional analysis of mass^-1 from the Newton and the (m/kg)^2. f(r) is a reduced form of the E/F ratio. the ~1.3x10^27 might be itself a measurement of "1" by balancing out the ratio of the universal physical laws of flat space (Special Relativity and Newton's Law of Universal Gravitation). In sum, perhaps this relationship shows that "1" is physically measurable, and demonstrates how mathematics (especially analysis) is intertwined with nature. – user149553 Oct 1 '17 at 21:11
  • Okay, so your f(r)=c^2r/Gm is indeed dimensionless. But A=mtr (t~secs) is >>not<< dimensionless, whereby A-->1 is meaningless. And work~energy=force*distance just looks like you're saying there's a mass-equivalent for gravitational energy (gravitational force times distance). So what? I'm not seeing any "1" dropping out of any of these relationships anywhere. And I'm double-positive I'm right about that -- not because I'm so smart, but because your little algebraic relations are simple enough so that about a million-zillion people would have already seen such an enormous coincidence long ago. – John Forkosh Oct 2 '17 at 4:23
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    I concede you are right about A = mtr. It should be just r --> 1. Thank you for the catch. The "1" drops out of dividing each side of the equation by r at the end of the algebra for f(r) = c^2r/Gm. ... I think the coincidence is hard to see because it involves the philosophy of the limit. Also the math is hard. We've been working 3+ years on the analysis and this is the first time acknowledging that it should be r-->1, not (A=mtr)-->1. There is something hard about seeing ~1.3x10^27 = 1 as correct and significant; this is why the limit notation should clarify some ambiguities. – user149553 Oct 2 '17 at 7:52

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