Georg Cantor

Really?  Well, not as far as I know - unless this tagger had proven something that would really shake up the mathematics community!

Here is the correct equation - and Cantor and me - at "The Cube" in Halle, Germany, where one side is dedicated to my favorite mathematician, Georg Cantor (1845-1918).

 

Cantor is my favorite mathematician, in part, because of his work with the infinite.  Before continuing with this post I have a question for you.  First, I want you to think about the counting numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and so on.  Now imagine removing all the odd numbers, so that you are left with only even numbers: 2, 4, 6, 8, 10, 12, and so on.

I said "and so on" at the end of each list, because each list is infinite; neither of them end.  But to get the even numbers you removed half the numbers from the first list - so are there fewer numbers in the second list because half were removed?  Or are there the same amount of numbers in both lists because they are both infinite, and infinity is just infinity?

SPOILER ALERT: Before the end of the post I will be giving away the answer as well as other related answers, so if you want to think about it on your own first, maybe move on to another post and come back to this one later.

Below are some photos of Cantor's home on 13 Handel Strasse in Halle.  It is now part of a row of houses, but when he moved to Halle and had it built it was built in a "green area," so it was one of the first homes and had yard space around it.

Cantor did very original work in mathematics, pioneering set theory and the theory of transfinite numbers.  Bold new work often brings out both admiration and anger.  Cantor's former teacher and mentor Leopold Kronecker, who, as it turns out, had a very different philosophy of mathematics than Cantor was appalled by his work.  The forms that Kronecker's responses took felt like persecution to Cantor.

Some other extremely prominent mathematicians of the day were also very negative towards his work.  One of these was Henri Poincare, who referred to Cantor's work as a "grave malady" from which someday mathematics would be cured.  Kronecker called Cantor a "renegade," a "scientific charlatan," and a "corrupter of the youth."

Others were supportive and more!  David Hilbert said, "No one shall expel us from the Paradise that Cantor has created."

Cantor felt persecuted by the negative responses (and issues of not being allowed to publish certain articles in certain journals as well as not being able to get a position in Berlin, which he really desired).  I'm compressing a lot of biography here, but Cantor also got hit with a number of personal traumas - the death of his mother in 1896, and the death of his son in 1899.  Additionally, it turns out that one of the main problems Cantor was working on (the Continuum Hypothesis) turned out to be impossible to prove (something that wasn't shown until well after his death).  What I'm leading up to in writing all this is that Cantor ended up struggling with bouts of depression.  It is very likely that he had bipolar disorder; recent scholarship bears this out, but this wasn't a diagnosis he was given during his struggle in the early 1900s.  He ended up spending time on and off in the sanatorium or "nervenklinik" - eventually dying there - of a heart-attack - in 1918.  (I'm sure it didn't help that leading up to this it had been hard to get nutritious food; this was in Germany during the time of World War I.)

Below are some pictures of the "nervenklinik" where he was treated and where he died during his last stay there.  It has been refurbished inside and out, of course, since 1918, but this is the same location and same hospital.

So, have you been thinking about that question about the size of the set of counting numbers and the size of the set of even numbers?

WARNING - ANSWER - The sets are the same size.  This can be shown pretty easily by using the common sense idea of "one-to-one correspondence."  For instance, I can put my hands together and match up all my fingers: thumb to thumb, pointer to pointer, all the way to pink to pinky.  This is one way that I can show I have the same number of fingers on each hand.  We can also match up the counting numbers with the even numbers.  If you apply a rule to every counting number - the rule "times two" you get an even number.  If you apply the rule "divided by two" to ever even number you get a counting number.  So for every counting number there is a unique even number that matches it (its double!), and for every even number there is a unique counting number that matches it (its half!).

That was a pretty quick explanation.  If you're interested and would like to talk further, please see me.  It's much easier in person!

One more question, but I won't make you think about it too long: what about the size of the set of counting numbers compared to the size of the set of fractions?  There are infinitely many fractions in-between every counting number, so shouldn't there be more fractions than counting numbers?  Here too Cantor was able to show that the sets are the same size.  (His proof is represented abstractly by the image with the circles and arrows on "The Cube" pictured at the top of this post and below.)

So now it sounds like infinity is just infinity and that you can't have bigger and smaller ones.

Well, it turns out there are infinities bigger than the infinity of the counting numbers.  

I'll leave it at that and let those of you who want to look into it further do so.  There are excellent books on this topic - and thousands of pages on the internet - and, if you know me, please come up to me and ask.  I'd be MORE than happy to explain it all :-)

As you can imagine, though, since Cantor was pioneering these ideas and was being harassed to some degree and feeling persecuted, that combination probably triggered some of his episodes of depression.  Galileo is someone else who had contemplated issues of size with regard to infinite sets.  He did so in a work that was published in 1638, and his conclusion was that terms like "greater than," "less than," and even "equal to" could not be applied to what we now call infinite sets.  The great Galileo could not get to the point that Cantor did, but Cantor boldly pressed forward, even against opposition.  At some point I need to go back and verify the following, but I recall having read something about Galileo saying that the infinite was beyond the realm of humans and that we needed to leave it to God.  Well, Cantor went to that realm and gave it to humans.

Cantor's tombstone - along with other family members - 

The stones on either side are for relatives of Cantor

Think what you will of Cantor's infinities (though as you do so, do be aware that these ideas are part of standard mathematics and that his work with set theory is now part of what is called "foundations of mathematics").  As for me, I'm with Hilbert:

"No one shall expel us from the Paradise that Cantor has created."

David Hilbert

I had one of the best experiences of my entire sabbatical today thanks to David Hilbert and a choice he made just over a hundred years ago! Soon after the turn of the twentieth century, physicist Max Born arrived in Gottingen, and mathematicians David Hilbert and Hermann Minkowski took Born, along with their families, on a walk up to Plesse Castle for a picnic.

After having climbed there yesterday myself I have a great deal of respect for how hardy they were! It's 8 km outside of Gottingen. I took a bus the first 6 km and walked the last 2 km, and there were points where the hill was so steep that I wasn't sure if it was possible to make the walk! Once I got to the top, however, I felt it was well worth the effort. The castle itself is fascinating, and the views of the surrounding countryside and the small village below are gorgeous. The vegetation around the area is just beautiful - wooded and filled with wildflowers. The air is so fresh. It was also refreshing to visit such a place and find it more visited by locals than by tourists.

I have no idea how much or how little time Hilbert spent here over the course of his years in Gottingen, but because it was one of my favorite things of the entire sabbatical trip and because I think it gives at least some insight into his life aside from his mathematics I'm going to post many pictures of this site!

Some folks took horses up. That was smart!

Clearly many other people decided this would be a good destination - as did I - as did Hilbert, Minkowski and Born a century ago.

 I enjoyed some time to journal here.

 One can also see Gauss's tower from here - one vertex of his triangle. (See this link for details.)

The afternoon here with Hilbert and Minkowski is one that Max Born never forgot.  What he writes of it later gives some insight into Hilbert and Minkowski:

The conversation of the two friends was an intellectual fireworks display.  Full of wit and humor and still also of deep seriousness.  I myself had grown up in an atmosphere to which spirited discussion and criticism of traditional values was in no way foreign; my father's friends, most of them medical researchers like himself, loved lively free conversation; but doctors are closer to everyday life and as human beings simpler than mathematicians, whose brains work in the sphere of highest abstraction.  In any case, I still had not heard such frank, independent, free-ranging criticism of all possible proceedings of science, art, politics.

I wouldn't be surprised if Hilbert had often hiked to the castle.  He was known to enjoy walking and hiking.  He would often walk with students or colleagues in the Gottingen Forest to the east of town, pictured below.

These woods were only about one kilometer from his house at 29 Wilhelm Weber Strasse.

When Hilbert began teaching at Gottingen in 1895 all classes were held in the same building, the Auditorienhaus, but eventually a plan would be developed to create a separate mathematics institute within the university with a building of its own.

The Auditorienhaus of Gottingen University

Prior to the building of the Mathematical Institute, the third floor of the Auditorienhaus was the center of mathematical life.  Here Felix Klein, who was instrumental in bringing Hilbert to Gottingen, established a mathematical reading room called the Lesezimmer, which was entirely different from any other mathematical library in existence at the time.  Books were readily available to students - on open shelves.  Additionally Klein established a collection of mathematical models that were housed here as well.

The Mathematical Institute began with a dream of Klein's but would be brought to fruition by Richard Courant and and David Hilbert.  It was opened in 1927.  It is a T-shaped building, and on the second floor at the top of the T, in a space built specially for it, resides the collection of mathematical models and instruments.

Gottingen Mathematical Institute from the back

Gottingen Mathematical Institute from the front

Upon entering the Mathematical Institute, one sees a study area for student use - and if one looks closely, one can see that the students have a sense of humor about their august mathematical heritage.

Bust of Hilbert in the Gottingen Mathematical Institute - as decorated by students

Directly above this study area is the mathematical models collection - of which more at this link.

David Hilbert and French mathematician Henri Poincare were the greatest mathematicians of their time. When Poincare died, Hilbert stood alone as the mathematical giant of his era.  It was during the time of Klein and Hilbert that Gottingen became the most important mathematical center in the world.

David Hilbert truly was a mathematical giant.  He would dominate one branch of mathematics, creating and publishing cutting-edge work in that branch and speaking of no other area of mathematics.  But then he would suddenly enter a new area of mathematics and produce in it immediately great mature work, and before long he would move on to another entirely new area of mathematics.

On August 8, 1900, David Hilbert gave a talk at the International Congress of Mathematics in Paris, which was held at the Sorbonne.  In his talk (and the proceedings) he presented 23 problems that he felt were the most important to be solved in the twentieth century.  This single talk definintely had a powerful influence on the direction of 20th century mathematics.

The opening words of his talk are given below, along with pictures of the Sorbonne.

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be towards which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

 In January of 1930, David Hilbert reached the mandatory retirement age of 68.  He delivered a "Farewell to Teaching" speech to a lecture hall crowded with students and other professors.  He was also honored by having a street (strasse) named after him.  Upon finding out about his honor, Hilbert's wife Kathe exclaimed, "A street named after you!  Isn't that a nice idea, David?"  It is reported that David shrugged and said, "The idea, no, the execution, ah -- that is nice, Klein had to wait until he was dead to have a street named after him!"

Sadly, towards the end of Hilbert's life his country entered a very difficult time with the rise of Hitler as chancellor in 1933.  Because of Hitler's policies, many professors, especially those who were Jewish but others as well, lost their positions at universities.  Gottingen had been the mathematical center of the world, and its Mathematisches Institut had only just opened its doors in 1927, but by 1933 when Hilbert was asked how mathematics was in Gottingen now that it had been freed from its Jewish influence, Hilbert, sadly, had to reply: "Mathematics in Gottingen?  There is really none anymore."

Hilbert died in 1943, but before visiting his grave I wish to visit one more topic of importance to him, so important to him that is engraved on his tombstone.  In the mid-to-late nineteenth century there was a widely embraced reaction against the then long-standing positivism of the explanatory power of science, and that reaction was "ignoramus et ignorabimus," that is "we do not know and will not know."  It expressed a position on the limits of scientific knowledge.  Both Hilbert and Minkowski found this idea abhorrent, and in his address at the Mathematical Congress in August of 1900 at the Sorbonne in Paris he expressed that mathematical knowledge is possible with human effort.  He said:

"We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus.  For there is no ignorabimus, and in my opinion none whatever natural science.  In opposition to the foolish ignorabimus our slogan shall be Wir müssen wissen — wir werden wissen! (Our motive must be to learn -- We shall this way greatly achieve!)"

Wir müssen wissen — wir werden wissen!

We must know -- we will know!

I found the tombstone, and the engraving (though, sadly, I see that it is fading) in Gottingen's Stadtfriedhof Cemetery -

A recording of Hilbert's voice exists and can be heard on this YouTube clip. This is from a radio broadcast he gave later in life in which he reiterated his positive credo, Wir müssen wissen wir werden wissen!, which can be heard at the very end (3:56-4:01)

Wir müssen wissen wir werden wissen!

We must know; we will know!