My Perspective on Mathematics

6/1/18

In my view, mathematics is the comprehensive effort to determine what we can say for sure using pure thought alone. In addition to the remarkable results achieved so far, mathematics can be characterized by two major errors, both introduced in antiquity. The first has been corrected; the second needs to be.

The first error was introduced by Euclid in his definition of the term 'axiom'. To Euclid, 'axiom' meant a statement whose truth was obvious because it was an incontrovertible statement about reality. By contrast, statements whose truth was not certain were called Postulates and were considered to be merely hypotheses.

This error persisted until Lobachevsky, Minkowsky, and Einstein, among others realized that the universe was not necessarily Euclidean after all, and that there were other geometries that might work better. Mathematics has successfully moved beyond that error by including non-Euclidean geometries and by foregoing any and all claims about reality.

The second error still contaminates Mathematics. That error is the acceptance of the notion of infinity. The notion was held suspect by such ancients as Zeno, but it persisted until it was put on a firm rigorous foundation by Cantor at the turn of the 20th century.

Cantor almost immediately recognized a problem of inconsistency, which is anathema in mathematics, and he dubbed it "Cantor's Paradox". Kronecker also noticed the error and fought against the acceptance of Cantor's theory into mathematics. He was overruled, however, by Hilbert who saw Cantor's work as a valuable and important contribution to mathematics.

Various people, including Frege and Russell worked on somehow removing or covering up the "antinomies" or inconsistencies engendered by Cantor's theory. A commonly accepted solution was Russell's Theory of Types.

Then, in 1931 Gödel proved the second of his incompleteness theorems and proved that no system of mathematics can be both complete and consistent. In spite of the attention given to this result, I believe the significance of the theorems has been understated. I'll explain why.

There is a trojan horse in one of Gödel's assumptions that is overlooked by everyone I am aware of. That is the condition that the system of mathematics under consideration must be robust enough to include arithmetic. That seems innocuous enough, after all, the set of natural numbers along with addition and multiplication seems to be a given. The trojan horse is the tacit assumption that the set of natural numbers is infinite. And, making this assumption brings along with it the inconsistencies mentioned earlier.

So, what is the problem with inconsistencies? Well, the story goes, if you could prove, for example that 1=2, then all bets are off and all mathematical results would be useless. But that is exactly what happened, without acknowledging the consequences. The Banach-Tarski Theorem demonstrates exactly this result: 1=2, and nobody seems to care. But I do.

As I see it, mathematics stands at the threshold of correcting this second problem, just as it did when it corrected the Euclidean problem. It is time to develop an alternative mathematical system that excludes the notion of infinity.

I predict that the alternative system will differ significantly in many respects from conventional mathematics that will seem to be important to mathematicians, such as the loss of real numbers, open sets, and continuity. But with respect to computation, it will be unchanged and thus just as useful to science and technology as the current mathematics has been. And, I predict that it will be even more useful.

It seems clear from the physical sciences that our reality is at basis grainy, or "quantized", instead of being continuous. So, a mathematics that is also grainy and discrete, should prove to be better suited to describing physical phenomena than the "old" continuous mathematics. So, to all budding young mathematicians in training, I ask you to put this under your bonnet and get working on it.

Please send me an email with your comments.

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