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 Why is math so adept at describing the world?
 Posted 2015-02-17, 10:40 AM Why is math so adept at describing the physical world? Except for some mathematical concepts which were developed explicitly for the purpose of describing the physical world, many mathematical concepts were developed without any forethought to their application. Yet years down the road they often find application in physics, and their usefulness in discovering new phenomena and accuracy in making predictions is astounding. This is inexplicable and marvelous to me. Why should the universe speak a language invented by us? Regarding this happy coincidence, the Nobel-prize winning physicist Eugene Wigner remarked "It is difficult to avoid the impression that a miracle confronts us here . . . ." It seems difficult to disagree with this without a sufficient explanation for why mathematics is so ridiculously good at describing the physical world. Thoughts?
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Demosthenes

Posted 2015-02-17, 01:33 PM in reply to Demosthenes's post "Why is math so adept at describing the..."
because Platoism.

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Last edited by D3V; 2015-02-17 at 01:36 PM.
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D3V

Posted 2015-02-18, 12:37 PM in reply to Demosthenes's post "Why is math so adept at describing the..."
Quote:
 Except for some mathematical concepts which were developed explicitly for the purpose of describing the physical world, many mathematical concepts were developed without any forethought to their application.
Can you be more explicit. A blanket statement saying that "many" were developed without any forethought is really wrong sounding. As "many" were developed for real-world application. Having relationships to more advanced formulas is no coincidence. You should go read up on the history of math.

Quote:
 Why should the universe speak a language invented by us?
It speaks a language that any intelligent race could conceive. I would not consider math to be invented by humans. It was more like discovery through our interpretation using symbols and signs.

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Goodlookinguy

Posted 2015-02-18, 05:56 PM in reply to Goodlookinguy's post starting "Can you be more explicit. A blanket..."
Goodlookinguy said: [Goto]
 Can you be more explicit. A blanket statement saying that "many" were developed without any forethought is really wrong sounding. As "many" were developed for real-world application. Having relationships to more advanced formulas is no coincidence. You should go read up on the history of math.
Sure, I can be more specific. A few examples:
• Conic sections were studied purely for their mathematical properties by the ancient Greeks almost 2000 years before they found an application in celestial mechanics
• Non-Euclidean geometry when studied by Riemann was considered to be mental masturbation until Einstein used it in General Relativity
• Euler's constant was studied as a problem of compounding interest until it found a use in nuclear decay
• The distribution of prime numbers has interested mathematicians for centuries, but only relatively recently found application in cryptography.

Even the famous physicist David Hilbert was astounded by the fact that math was almost unbelievably applicable to the real world. In fact, he was astounded by how this fact applied to even his own work: "I developed my theory of infinitely many variables from purely mathematical interests and even called it 'spectral analysis' without any pressentiment that it would later find an application to the actual spectrum of physics."

It seems to me that much of the mathematical development, especially of the previous two centuries, has occurred without consideration for real world application.
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Demosthenes

Posted 2015-02-18, 07:48 PM in reply to Goodlookinguy's post starting "Can you be more explicit. A blanket..."
Goodlookinguy said: [Goto]
 It speaks a language that any intelligent race could conceive. I would not consider math to be invented by humans. It was more like discovery through our interpretation using symbols and signs.

That reminds me of Max Tegmark's mathematical universe hypothesis, which posits that the universe is a mathematical structure, and that mathematicians are akin to archaeologists uncovering hidden or forgotten truths. Perhaps it's just semantics to attempt to distinguish this from some form of "true" creativity, but I like the idea that even our most obscure mathematical theories can't be completely decoupled from physical reality.

He goes further to state the Computable Universe Hypothesis, which "posits that all computable mathematical structures (in Gödel's sense) exist". Consequently, perhaps the existence of absurdly large, computable numbers (see this for some examples) that are seemingly too large to describe anything in our own universe hints at the existence of a multiverse where they may find some use. (That said, some disturbingly large numbers seem to emerge from physical considerations in combinatorics from time to time...)

As a counterargument to the idea that math is adept at describing the world, there are quite a few elegant structures that "should" be exact, but are actually approximations. For instance,

$\reverse \left(\frac{1}{10^5}\sum_{n=-\infty}^{\infty} e^{-\frac{n^2}{10^{10}}}\right)^2\approx\pi$

is correct to at least 42 billion decimal places (but is provably not equal to $\reverse \pi$).
"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram
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Chruser

 Posted 2015-03-02, 10:37 AM in reply to Chruser's post starting "That reminds me of Max Tegmark's..." I was actually working on a problem yesterday that reminded me of your comment, GLG. In Statistical Mechanics you often encounter problems where energy surfaces in phase space are higher dimensional hyperspheres. Doing mathematics on this surface requires some knowledge of the volume and surface area elements of these hyperspheres. The mathematics of such higher dimensional geometries was worked out well before the modern formulation of Statistical Mechanics, or even of Hamiltonian mechanics (which is a reformulation of Newtonian mechanics upon which Statistical Mechanics depends).
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Demosthenes

Posted 2015-03-03, 07:04 AM in reply to D3V's post starting "because Platoism."
D3V said: [Goto]
 because Platoism.
I'm getting really into Platoism lately.
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Skurai

Posted 2015-03-09, 11:46 AM in reply to Skurai's post starting "I'm getting really into Platoism lately."
Skurai said: [Goto]
 I'm getting really into Platoism lately.
And...?
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Demosthenes

Posted 2015-03-18, 01:18 PM in reply to Demosthenes's post starting "And...?"
Demosthenes said: [Goto]
 And...?
Nietzsche was a bitch.
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Skurai

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