It would be great if your example was from a respectable philosopher but anything will do. currently I am writing a debate case based on the resolution of "rationalism should be valued over empiricism" and I can not find and example of empirical math anywhere.
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The question is ambiguous between, "A math equation applied to empirical phenomena, and, "A math equation that can be proved by empirical experiments." Even things like the four-color theorem are taken to be justified a priori (on some working definition of apriority, at least), but perhaps this depends on gerrymandered definitions of pure reason vs. experience? – Kristian Berry Nov 10 '23 at 23:30
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It's not clear what you mean by "an empirical math equation". Newton's law of gravity is a math equation that was derived in part through observation. – David Gudeman Nov 10 '23 at 23:39
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Related: Fundamental equations in Philosophy and Logic. – Weather Vane Nov 10 '23 at 23:40
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Hooke's law was an equation inferred from experiments. Or, perhaps, you mean purely mathematical equation that was discovered empirically? Bifurcation rate formulas with Feigenbaum constants would be an example. Mill argued that geometry and arithmetic were empirically surmised. There is a whole field of Experimental mathematics now. – Conifold Nov 10 '23 at 23:53
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y = xz F = ma (Newton's
law). Quid vide? Do nfr things. – Agent Smith Nov 11 '23 at 16:10
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Not sure why a philosopher would be doing "empirical math" but engineers have tons and tons of empirical equations, and science uses math-assisted empiricism all the time.
Examples include most of classical thermodynamics, hydraulics, aeronautics, just to name few. Also, most of our core mathematical concepts are grounded in intuitions gained from experience: groups of things, numbers, points, lines, spheres etc.
So I'd actually like to see purely rational math without a hint of empirical influence :)
EDIT:
@WeatherVane pointed out that imaginary numbers arose outside of any empirical need (at the time!) so would be a good answer to my question. Thanks!
Annika
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1Complex math involves the square root of -1. This is purely rational, as it cannot exist in the real world. – Weather Vane Nov 11 '23 at 20:34
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@WeatherVane so are you saying imaginary numbers are not empirically motivated? – Annika Nov 11 '23 at 20:36
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1Yes: Wikipedia's page Imaginary number says: "Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless . . .". That's hardly an empirical motivation. – Weather Vane Nov 11 '23 at 20:49
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@WeatherVane +1 got it. that does seem to fit the bill. They were invented to fill gaps in the solutions to algebraic equations, which does seem to be a purely rational exercise. They do find many many uses LATER ON, but that is beside the point. Will update my post :) – Annika Nov 11 '23 at 20:57
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@WeatherVane I'm confused, are you taking issue with me acknowledging your input? Or are you saying I should have used a "?" above to make it properly a question? I used it more as an indirect question. – Annika Nov 11 '23 at 21:18
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1Your answer is correct though: much of science derives from observation. One exeception is the Higgs boson, with was predicted theorectically. – Weather Vane Nov 11 '23 at 21:25
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@WeatherVane complex numbers are used all the time in physics. They represent rotations in 2-dimensional orthonormal mathematical spaces. – g s Nov 11 '23 at 22:09
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@gs I took WeatherVane's response to mean that the genesis of complex numbers are not empirical, but came from a purely rational motivation. – Annika Nov 11 '23 at 22:10
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@Annika you're looking for abstract mathematics, the (incomplete) reformulation of mathematical principles from bottom-up logical principles. You'll want linear algebra and differential equations as prerequisites. – g s Nov 11 '23 at 22:11
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1@gs but they are not empirical, since imaginary numbers cannot be observed. I did not say they have no use. They are a useful stepping stone to get from A to B. But practical does not mean empirical. – Weather Vane Nov 11 '23 at 22:29
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Some examples of empirical mathematics are in
10 Equations to Expand Your Macroeconomics Expertise
One of them is
Phillips curve relationship
The Phillips curve relationship says that in the short run, a negative relationship exists between inflation today (πt) and unemployment today (ut). Also, when actual inflation today (πt) equals expected inflation (πe), unemployment is equal to its natural rate (un).
So in the short run, reducing unemployment below its natural rate is possible, so long as you can surprise people by creating higher inflation than they expect.
Another example is from The Astrophyical Journal
An Empirically Derived Formula for the Shape of Planet-induced Gaps in Protoplanetary Disks
. . .
This new analytical formula . . .
Weather Vane
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