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Certainty in Mathematics September 2015

Today I will be discussing Certainty in Mathematics.

Pythagoras.

The Plimpton 322 tablet is believed to have originated in 1800 BC Babylonian. It is written in Cuneiform script.

Scholars are not in full-agreement of the translation/interpretation, some speculations are that these numbers represent Pythagorean triples, numbers that can be related by a^2+b^2=c^2

You have probably learned the theorem which is attributed to the Pythagoreans, that the sum of the squares of the lengths of two legs of a right triangle is equal to the square of the length of the hypotenuse. However, did Pythagoras or his followers actually invent this important Theorem?

Figure: Medieval woodcut

Andrew Dynneson University of Texas at El Paso [email protected]

Figure:Raphael painting of the Pythagorean School


Pythagoras (6th Century BC) was a Greek Philosopher. What you may not have learned in school was that Pythagoras founded a religious cult which was based on the ubiquity of numbers in the universe (for example, music and the movement of the heavenly bodies). Among other things, the Pythagorean School was vegetarian.

Unfortunately very little factual information is known about the cult during the time Pythagoras was alive. Fragmentary and disagreeing accounts that were not written until 150 years later are the earliest surviving record. The more organized manuscripts of later Pythagoreans on the life of Pythagoras were written two centuries after-the-fact. I will be commenting somewhat on Iamlichus’s On the Pythagorean Life, since I perused some of these pages, translated into English, as an undergrad.

( http://plato.stanford.edu/entries/pythagoras/#PytQue )

There is a folk-lore going around among mathematicians that Pythagoras was so enamored with whole numbers and their ratios, and at the same time the group was well aware of the existence of the existence of √2, since the theorem that takes his name clearly indicated this existence.

However Hippasus challenged the Pythagorean world-view that “All is number,” by showing that √2 could not be represented as a ratio of numbers (we now call these ratios a/b “rational” numbers, and the “numbers” of the Pythagoreans “integer” numbers).


Figure : The History of Philosophy (c.1660) by Thomas Stanley.

http://plato.stanford.edu/entries/pythagoras/#PytQue


At any rate, the legend goes that the early Pythagoreans were so incensed that anyone would dare challenge the universal ubiquity of numbers, that Hippasus was executed by being thrown off a boat into the ocean.

This account is likely fictional. Although there is some mention of it in Iamblichus, it is not stated directly that Hippasus was executed, merely that he “perished at sea for his impiety”. If he had drowned, it is possible that the neo-Pythagoreans had viewed this as some sort of divine-retribution. It is also likely that he had not actually been working with the square-root of two, but rather some other “incommensurable” ratio

(http://www.science20.com/beamlines/was_hippasus_pushed_and_other_mysteries_of_mathematics).

Uncertainty.

I am spreading this misinformation since this fable is indicative of the common thread of this lecture. Namely, that Mathematics tends to have a mystique of certainty. That to deny some aspect of its claims is illogical (“irrational” numbers for example [joke]). On the other hand, it is clear that math does undergo changes as it evolves. The mathematics of today are very different to the mathematics of yesterday. The Pythagorean cult may have been offended by the modern offering of a “real number,” and balked at the claims of the “imaginary number,” √-1 . Today, including these numbers in equations, without any reservations on their existence, is common-place. And, the fact that a number is irrational or imaginary seems to have no bearing on their restraint; they are utilized liberally.

My opinion is that math is often very good at absorbing new information. Numbers is one example. The incorporation of Chaos into mathematics while still maintaining the results from Determinism was visited in the previous lecture. However, with change there is often turbulence, as the certainty so sought-after by mathematicians is lost again and again throughout history. The new information is absorbed, digested, and repackaged. What you see when you take a course in math is the finished product. What is invisible at that end-stage are the struggles that the mathematicians underwent to come to those conclusions, the institutional biases that initially resisted the proposed changes, and other such controversies.

Regarding the repackaging of concepts, and the certainty that is displayed for public-consumption:

“Without it, the myths would lose much of their aura. If mathematics were presented in the same style in which it is created, few would believe in its universality, unity, certainty, or objectivity” (Hersh, 1991, p. 130–131).


Figure:Hippasus

http://www.science20.com/beamlines/was_hippasus_pushed_and_other_mysteries_of_mathematics


Hilbert.

David Hilbert was a 19th and 20th Century German Mathematician and Philosopher. In 1900, the International Congress of Mathematicians convened in Paris. It was here that Hilbert presented a list of unsolved problems that Hilbert considered important. These problems would have a profound influence on the development of math in our modern time.

Hilbert was familiar with many aspects of mathematics, and was considered a “universal mathematician.” He also worked extensively on Physics. He is also credited as being one of the founders of modern rigorous mathematical logic, although classical logic has been around since Aristotle.

He often relied on the Law of Excluded Middle in his proofs, as well as solving problems in indirect ways, by proving that something exists without constructing the object directly. It was almost as though the methods of proof that he developed were

more important than the actual problem itself. When he submitted his proof of Gordon’s Problem, Gordon rejected Hilbert’s proof, “This is not Mathematics. This is Theology.” However, Klein intervened and pushed the article-through. When Hilbert expanded his methods in an additional article Klein remarked, “Without doubt this is the most important work on general algebra that the Annalen has ever published.”

The Law of Excluded Middle is simply that a proposition is either true or that its negation is true. In Symbols:

“p or not p is true.” This shows that foundational logic is consumed with binary thinking. A proposition is either true or false. It cannot be both true and false at same time. If something is false, simply negate it and its negation must be true.

This foundational logic quickly becomes meta-mathematical in nature, it is a logic which refers to itself.

Russel is sometimes quoted as presenting this paradox: Let A be the set which contains all sets that do not contain themselves as elements. So then A containing itself goes into a sort of flux, where it neither contains itself, nor does not contain itself.

Similarly, p= “This statement is False,” is a proposition which refers to itself, and seems to violate the excluded middle since it is neither true nor false, but some other paradoxical sort of truth.



Brouwer.

Kronecker was an early critic of Hilbert’s methods, and clung to a more constructivist philosophy. Brouwer picked up the torch and founded an “intuitionist school.”

In undergraduate courses like this one, the question that arises is cliché. Is mathematics discovered or invented? Intuitionism essentially believes that mathematics is a human-invented concept, which generates more and more layers of complexity, but only as a mental construct. Mathematical Truth is in-effect has a purely subjective existence. In this way, it can generate a paradox and still be a valid way of exploring symbolic meanings, since all you have created are shadows of the concepts they were thought to represent.

Where Plato and Hilbert would have argued for a more certain, objective truth, Brouwer shed the reliance on the excluded middle. A proposition can be true at one time, and false at another time. For example, “The world is flat,” would never have been disputed at some point in time. However, at present time it would be equally foolish to deny its negation, “The world is not flat.”

While Intuitionism has its charm, I would not consider myself an Intuitionist according to this definition perse, since I like to think that the work I am doing is indicative of something meaningful. However, I am forced to concede that paradoxes arise within the symbolic forms that may not be resolvable. Can something be some superposition of truth and falsehood? Perhaps there are “fifty-shades” of truth, or a continuous-fuzzy sort of truth, or in the example of the statement which refers to itself as False, which is neither true nor false, but a “Flux-True/False,” or something along those lines.

While it is possible that logic does not represent anything in reality, it is also possible that the paradoxes of logic are really indicative of the paradoxical nature of reality, however this thought is once again going beyond the scope of this lecture.

Rather, I tend to borrow from all sides of this ongoing discussion, as I borrow from Platonism and Intuitionism alike in my work. I feel that this is actually pretty standard among career mathematicians of our time. I do feel that math should make “intuitive sense,” and that the intuition behind a concept is where there is beauty and art in mathematics. And there is a sort of ad hoc nature to modern mathematics. There is a sort of, “Let us try this, and see what happens.” However, the ubiquity is realized and explored by some.

Intuitionism represents for me a sort of freedom to explore contradicting view-points in mathematics. Perhaps there is some new, other kind of emerging philosophy of mathematics that is not Intuitionism, nor Platonism, but something entirely new.


Brouwer Photograph


Gödel.

Gödel represents, ultimately, the death of certainty in modern mathematics. He proved the Incompleteness Theorem. He proved that within any formal theory that contains arithmetic truths, there exists an arithmetical statement that is both true and unprovable.

This put an end to Hilbert’s program in particular, who wanted to prove by way of his meta-mathematics that “all of mathematics follows from a correctly chosen finite system of axioms.”

Gödel proved that this is ultimately absurd, and proved with sufficient generality that no such system exists.

It is my own personal conclusion, these people that are mentioned here essentially built and destroyed mathematics, and we are now living in the after-math. We have all of these incredible abstract tools to draw upon, while at the same time, certainty is imposed or illusory.

Just because Hilbert’s program was obliterated does not mean that theoretical mathematics is doomed. It will absorb this new information, and continue on its merry-way. It simply means that our over-reliance on this system should be re-examined.1

1 For further reading, see Kline: Loss of Certainty.



Abstract from Hofstadter: Gödel, Escher, Bach.


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