Pythagoras' Legacy by Danesi Marcel;

Author:Danesi, Marcel; [Danesi, Marcel]
Language: eng
Format: epub
Publisher: Oxford University Press USA - OSO
Published: 2019-11-29T00:00:00+00:00

Epilogue

All of mathematics is encoded in symbolic notation. The Pythagorean equation c2 = a2 + b2, for example, is essentially a convenient notation of saying the same thing as the sentence “the square on the hypotenuse is equal to the sum of the squares on the other two sides.” In so doing, it takes the semantics in the linguistic sentence out, leaving only the structural (symbolic) outline of the information. It is this feature that makes it cognitively powerful, since we can now find many more meanings and applications for it, in addition to the original geometrical one. For example, one can now ask what integers fit the equation and, further, if there are other exponents for which the equation holds in general, cn = an + bn. From this deliberation on the notation itself, detached from its original geometrical meaning, has come much subsequent mathematical contemplation leading to such intriguing ideas as Fermat’s Last Theorem, as we saw. In other words, notation suggests meanings that could not otherwise be contemplated.

The ancients did not have exponential notation at their disposal, so they used other devices to accomplish many of the same things as the new notation allowed so efficiently. In his book, The Sand Reckoner, Archimedes discussed the notion of powers in a roundabout way. However, it did not allow him to develop such techniques as exponential laws and logarithms.

Einstein’s equation E = mc2 has imprinted in it a lot of information about physical reality that could not be expressed in any other way. It says, in a nutshell, that the speed of light is constant and thus that it constrains physical reality. What happens if there is a universe where this formula does not hold? It would be unimaginable, even though it is possible. Above all else, we would need a notation to devise formulas for such a universe. As the philosopher Ludwig Wittgenstein (1922) put it, “Whereof one cannot speak, thereof one must be silent.” As the language of Nature, mathematics breaks the silence periodically. E = mc2 speaks volumes, to belabor the point somewhat.

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