Crackpot Spammer Ch 09
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Disclaimer: The following short story is fictional. No resemblance to any person, living or dead, is intended or should be inferred.
Faustus begins his lecture by discussing significant figures: ‘Consider a number of the form ‘1.234567,’ or thereabouts. Such a number clearly has seven significant digits. Such a number might arise from rounding the transcendental irrational number pi, for example. The number pi is given by 3.1415926535898…. Rounded to seven significant figures it is 3.141593. For almost any imaginable scientific or engineering endeavor this is a suitable approximation to pi. Likewise the number ‘e’ may be approximated by 2.718282 and the square roots of two and three are 1.4142134 and 1.732051. Many scientific calculators, e.g., the TI-30XIIS, display ten significant digits. For purely mathematical constants, as those just mentioned, this is both accurate and precise.’
Faustus pauses a minute to contemplate accuracy versus precision and then continues: ‘But for the real world there may be lurking (or exogenous) variables influencing the numerical display. These may affect the accuracy of the displayed number. There may also be spurious digits arising from less significant input values. Let’s investigate for a moment with the square root of three, rounded to seven significant figures. (George Washington was born in 1732.) 1.732051^2 = 3.000000667. This is rounded to 3.000000. This is the integer number three, more or less, mostly more. Notice how accurate the square of 1.732051 actually is!’
‘We will look at two physical constants and one mathematical constant. The mathematical constant is e^((pi^2)/2) = 139.0456366605…, which rounds at seven significant digits to 139.0456. This is a ‘bad news’ number for physicists, increasing the scope of pure mathematics at the expense of the pedants pushing extra digits to obtain advanced degrees. No small wonder that it is named after the actress known as ‘The Public Enemy of All Mankind.”
‘The two physical constants are the Electron g Factor and the Fine Structure Constant. Let’s first consider the Electron g Factor, since it is easier to discuss and has measurement considered more precise and accurate than the Fine Structure Constant (and its inverse). The most recent recommended value for the Electron g Factor is 1.0023193043622. Physicists claim this is a very good approximation, both accurate and precise. For the moment, let’s take them at their word. After all, it is a single, simple measurement, not an artifact constructed from several other constants as is the Fine Structure Constant and its inverse.’
‘Let us list the eleven values of the inverse Fine Structure Constant from learned bibliographies from 1969 to 2006, in increasing order: 137.03544, 137.03545, 137.03591, 137.03597, 137.0359895, 137.03599883, 137.0359997, 137.0360017, 137.0360017, 137.0360119, 137.036073. Now we have a range of values to look at. Let x denote the value obtained from the equation x + 1/x + 2.0023193043622 = e^(pi^2/2). It is 137.036020005 and 1/x = 0.0729735145521. It lies between the seventh and eighth estimate. Clearly this is a candidate, assuming that the Electron g Factor is correct. What can be done to ‘sharpen’ this estimate? It has been stated in several textbooks and mentioned in the physics forum that the Fine Structure Constant and the Electron g Factor are interdependent or somehow entangled. This could be one such functional relationship. Could this be ‘the’ functional relationship? If so, the Ansatz e^(pi^2/2) is a pure mathematical constant with physical significance.’
‘What remains to be done to firm up this theory? The Electron g Factor needs to have a pure mathematical formulation. This would require some special knowledge of the structure of the electron. The dimensionless physical constant needs some elementary formulation in terms of pi, e, the square root of an integer, and the imaginary number i.’
‘If History teaches us anything about the Fine Structure Constant, it’s that its value moves around, but its inverse is always near 137.0360. This matches with our estimate in seven significant digits. Now the challenge is to find a closed-form expression for the Electron g Factor. Accomplish this and the Fine Structure Constant will be proven to be a mathematical constant. Then one can ‘back engineer’ the value to the correct geometric model of fundamental physical particles. But this is a very difficult feat to accomplish.’
Faustus concludes: ‘It was the Seventeenth Century before the number pi entered into the domain of mathematics instead of physical measurement. And the scope of mathematics was expanded as irrational numbers were understood to be either algebraic or transcendental. While rational numbers and algebraic irrational numbers are countable, the transcendental irrational numbers are uncountable. And our understanding of the mysteries of mathematics was expanded.’
Summing up, Faustus says: ‘Likewise, should the Electron g Factor and the Fine Structure constant be found to be mathematical constants, Physics will lose one of its prized joys and mathematics will find new foundations. In fact, the understanding of the geometry of fundamental particles will be expanded. Rather than just tossing out some recipe or formula, like the Schrödinger wave equation, working models may be constructed. The present powers-that-be are diametrically opposed to such a situation. Having become aware of the constant e^(pi^2/2), it is time to determine the basis for the Electron g Factor.’
25 Jul 2011 Taunus Trumbo
Grace
- 06.08.2022
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