This statement is attributed to the mathematician Leopold Kronecker (1823-1891), who insisted on the constructibility of mathematical objects. He believed that all mathematics could be reduced to arguments using only the integers and finite numbers of operations. He was violently opposed to such things as the use of irrational numbers, transcendental numbers, upper and lower limits, and the Bolzano-Weierstrass theorem (well, much of the new mathematics being developed by Karl Weierstrass for that matter), as these devices he felt produced objects that did not exist. This extreme philosophical viewpoint on mathematics caused him to quarrel with many mathematicians, even going so far as to block publication of papers by Heinrich Heine (of the Heine-Borel theorem) on Fourier series and papers by Georg Cantor on transfinite numbers and set theory (not because he personally didn't like Cantor, as asserted by some of Cantor's biographers, but only because he was violently opposed to Cantor's ideas) in the influential Crelle's Journal. In 1889 Ferdinand von Lindemann produced a proof that π was transcendental, and Kronecker was said to have given von Lindemann the backhanded compliment: "Of what use is your beautiful proof, since π does not exist!"

This extreme point of view, which made Kronecker many enemies in his time, was actually a view first propounded by Pythagoras, who called the irrational numbers that he discovered to his consternation "unutterable" (this is the reason why the word surd is used to designate the irrational roots discovered by the Pythagoreans, it is ultimately derived from the Latin for "deaf-mute"). Leibniz himself spoke of the "labyrinth of the continuum" when referring to the philosophical troubles that the very idea of real numbers is fraught with. In fact the term "real number" is something of a misnomer, as they are actually quite unreal! In fact, it can be shown that almost all real numbers are transcendental, uncomputable, and cannot even be named! Mathematicians in the century after Kronecker managed to show all of these, and with these discoveries, Kronecker's position doesn't seem quite as untenable as it seemed to his contemporaries.

First of all, we start with the theory of real numbers that was proposed by Cantor and Richard Dedekind, which Kronecker was so vehemently opposed to. Dedekind managed to give a definition of a real number in terms of what are today known as Dedekind cuts, and Cantor managed to show that the real numbers are non-denumerable, that they are a higher-order infinity than the integers by using the diagonal argument that bears his name. Since the integers, rational numbers, and algebraic numbers are all denumerable, then that means that most real numbers are actually transcendental.

However, in the early twentieth century there began to appear intimations that there was something terribly wrong with the notion of a real number as it has been thus developed. Emile Borel in 1927 pointed out that if you consider a real number as an infinite sequence of digits then you could put an infinite amount of information into a single number. He came up with a number, known as Borel's constant, that could serve as an oracle to answer any yes/no question put to it. Today, Borel's argument might be stated a bit like this: let us treat each possible ASCII text as though it were a single number, for instance "Do real numbers exist?" would correspond to the hexadecimal number 0x446F207265616C206E786973743F, or 1,388,008,220,904,010,789,705,024,363,787,327 in decimal. Then we take, say, the 1,388,008,220,904,010,789,705,024,363,787,327th digit of Borel's constant in base 4. If the digit is 0, then the number does not correspond to a valid question, if it is 1 then the question is unanswerable (e.g. "Is the answer to this question 'no'?"), 2 if the answer is no, and 3 if the answer is yes. Such a "know it all" real number is certainly present in the set of all reals. But then Borel asks this troubling question: "Why should we believe in this real number that answers every possible yes/no question?" And he concludes that he doesn't believe it, there is no reason to believe it, that such a thing should exist is totally absurd!

In 1936, Alan Turing published his landmark paper "On computable numbers, with an application to the Entscheidungsproblem" where he arguably takes the first steps to inventing the computer. In this famous paper he found out, among other things, that most real numbers are actually uncomputable! He realized that just as with Borel's numbering of questions, it is possible to number all of his Turing machines, all computers (if the Church-Turing thesis is to be taken at face value), with an integer called a description number. This means that the "computable numbers" that Turing defines are denumerable, and hence most real numbers, which according to Cantor are non-denumerably infinite, must also be uncomputable! Turing himself explicitly shows a particular real number, Turing's constant, that is uncomputable in this sense, very much in the way that Borel constructs his number. Take the Turing machine with the description number, say 313,322,531,173,113,325,317 (my example in the description number node, using Turing's own universal Turing machine). If that machine halts, or doesn't correspond to a valid Turing machine (in which case we say that it corresponds to a TM with no states) then the 313,322,531,173,113,325,317th binary digit is 0, else it is 1. (In this particular case, that bit of Turing's constant must be 1, because the machine corresponding to that DN is one that alternates printing 1 and 0 on its tape forever.)

Also, one cannot even give names, one cannot even refer to most real numbers. If you fix the formal language or axiomatic system (which are really the same thing) you use to refer to your real numbers, then the set of all possible names or referents is still denumerable. Since you have an uncountable infinity of real numbers to name or refer to, then nearly all of them are not only transcendental and uncomputable, you can't even specify any of them uniquely, a specific real number of this type can't be defined or even singled out!

So Kronecker doesn't seem so crazy being against real numbers now, does he? Why should we believe real numbers are "real" if we can't even calculate them, we can't even give them names, or refer to them uniquely? And yes, this vast uncountable infinity of real numbers is even more unutterable than Pythagoras could have imagined.

Furthermore, real numbers seem even more untenable from a physical point of view. No physical constant has ever been measured to greater than 20 decimal places of accuracy. There have been many indications that the model of the universe as a continuous place where infinitesimally short distances and times exist is incorrect. For instance, Maxwell's equations show that the electric flux density at the location of a point charge like an electron is infinite! The only way that Richard Feynman could get around the infinities that appear in his formulation of quantum electrodynamics was to use the mathematically-dubious technique of renormalization, and one cannot help but think that Feynman may have obscured something fundamental with that cavalier approach to the mathematics. String theory on the other hand, was developed partly to eliminate the infinitesimal distances that caused such troubles in quantum field theories such as quantum electrodynamics (but unfortunately string theory as it exists today doesn't seem to have eliminated all of these infinities yet, though). According to quantum mechanics, infinitely precise measurement requires infinite energy, but long before that what you're measuring will undergo gravitational collapse. Quantum mechanics, indeed its very name, suggests that the universe is not continuous, but quantized. We have quantization of charge, we have quantization of matter, even the quantization of energy and angular momentum, why not quantization of space and time itself? Perhaps that's part of the reason why quantum gravity is so difficult.

So perhaps even when God created the universe, God only created the integers, and all else really is the work of man!


Leopold Kronecker biography at

Gregory Chaitin, "Meta Math!" at

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