Thus began a talk that, it can be argued, led to the invention of the general purpose computing device, or the computer, as we know it today.
This was a remarkable talk that was a powerful driving force over mathematics for all of the 20th century. This talk was given by the eminent mathematician David Hilbert, who at the time had already begun to get the reputation of being the greatest mathematician of his time, along with Poincare.
The talk was given in 1900, at the International Congress of Mathematicians, at Paris (as an aside, he didn't talk about all the problems at the venue, and a complete enumeration was only given later).
In his talk, titled, simply as "Mathematical Problems", he chose 23 problems that he considered would lead to results that would substantially expand on mathematics and at the firm up the underlying foundations of mathematics (one problem - a rather vaguely worded one - was on physics).
Now David Hilbert was a firm believer in the power of the intellect and human reasoning, and of the purity and rigor of mathematics. He was convinced that once set of a firm and logical foundation, every problem in mathematics could be solved. He said, of solutions to mathematical problems in general that, “You can find it by pure reason, for in mathematics there is ignoramibus". (The reference to ignoramibus is from the statement on the limits of scientific thought by Emil du Bois-Raymond, that "ignoramus et ignorabimus" which roughly translates to 'We do not know and we will not know')
I would highly recommend that anyone with any interest at all in mathematics would benefit greatly in reading at least the introductory part of his talk[1]. It is rare that one can see the beginning of a mathematical movement, the starting point of something that has dominated much of the scientific landscape of a century. While reading the text of the talk, one can almost pinpoint the statements that launched Godel and Turing into their landmark discoveries (or inventions - but that's the subject for another post).
Let's take a few of the problems and try and go a bit into them. The primary obstacle here, is that for most of us, our education of mathematical developments is restricted to approximately the 1850s or so, with any advances after that showing up on our radar screen only if it has direct relevance to the subject we specialized in. In my own case, it would be a limited knowledge of number theory (due to interests in cryptography), computer sciences or in engineering aspects of things like model-order reduction. So a lot of the mathematics of the 20th century is just too hard for us to get through (and it's not that it's that much more harder - it's just that the amount of foundational stuff you have to know to understand it is humongous).
Luckily, in the case of Hilbert's questions, some of the questions themselves are fairly understandable, even given the general limited knowledge of mathematics of the 20th C among the general populace (incl. self in this)
I'll take one of his questions in this issue, and some others in future editions. The second one, which I hope to tackle in the next issue, is the one that deals with the fundamentals of mathematics, and which was dealt a body blow by Godel, and than by Turing, and led indirectly to the concept of the general purpose computer.
I) Cantor's problem of the cardinal number of the continuum
This was the first problem that Hilbert mentioned. It deals with set theory and the concept of infinity.
To most people, infinity is an abstract representation of the largest natural number. But Georg Cantor showed that there is more than one level of infinity. The smallest infinity is "countable infinity", and the simplest example of a countable infinity is the cardinality of the set of natural numbers (this is what we generally tend to think of as infinity in elementary mathematics - the number of elements in the set of natural numbers "1, 2, 3,....."). This infinity is commonly called aleph0. Cantor stated that besides this, there are other "uncountable infinities". The cardinality
of the set of real numbers is an example of "uncountable infinity". So the number of real numbers is the first uncountable infinity. This is generally called aleph1.
Bear in mind that real numbers are used to represent the linear continuum (for an example of a continuum, take time. Any individual point in time is mathematically meaningless, and the only accurate representation would be of the continual flow from one point to another. Similarly Euclidean space is another example of the continuum).
Cantor's continuum hypothesis, stated simply, is that "aleph1", the cardinality of the set of real numbers, is the next highest infinity after "aleph0". Note that it is fairly easy to show that aleph1 = 2^aleph0.
Now set theory is defined in terms a set of axioms called the Zermelo-Fraenkel axioms. The ZF axioms, along with the axiom of choice, can be used to define all of ordinary mathematics (interested people can and should search the web for more details on the ZFC axioms). It is generally accepted that ZFC is consistent, though this is of course unprovable, because ZFC contains arithmetic, and Godel had already proved that for any axiomatic arithmetical system, consistency is non-provable.
In 1938 Kurt Godel proved that the ZFC axioms cannot be used to disprove the continuum hypothesis. In 1963, Paul Cohen (who won the Fields medal for this work) went on to prove that ZFC cannot be used to prove the continuum hypothesis.
So, in essence the continuum hypothesis is undecidable in set theory, at least as defined by the currently accepted axioms of ZFC (or even for all extensions that have been tried so far).
So even today Mathematicians have no real proof of the Continuum Hypothesis, and it has been shown that in existing formulations of set theory, it is not possible to prove (or disprove it). However, work is progressing on alternate mathematical frameworks where it may be possible to do so.
