Maxx Cho
This review on Amazon of an algebraic topology textbook is perhaps the funniest review I have ever read:
By the way, this textbook is available completely free directly from the author as a gigantic PDF file: http://www.math.cornell.edu/~hatcher/AT/ATpage.html
I personally prefer Munkres’s Algebraic Topology textbook, but the Munkres does not cover homotopy theory, and is not free (no pun intended).
If there is one area of mathematics I do not enjoy, it is combinatorics. Whenever people start saying things like “n choose k, but divide by two because you double counted,” it goes right over my head. I can’t quite put my finger on it, but there is something really hard about counting arguments for me.
I would have dismissed combinatorics outright as being petty and stupid (perhaps unfairly) if it had not been for my Topology seminar last semester. In studying simplicial homology, I learned that at the heart of space, there is something fundamentally combinatorial about it. You see, simplices (which are in essence n-dimensional generalizations of triangles) are defined and dealt with in a purely combinatorial fashion. And yet, this reveals something deep and fundamental about the nature of space.
The moral of the story is that combinatorics is very fundamental. It seems to crop up in nearly every branch of mathematics. And yet, it seems to be the only area that category theory has not swept under the rug as being a “trivial application” of category theory.
I wonder if there is a way to formalize and abstract counting arguments in a category-theoretic fashion? (I think Yoneda’s lemma is already hinting at this.)
P.S. Sorry about the exceptionally nerdy post. Not every post is like this, I promise.
I have observed something very curious about studying mathematics. Ideas that seem very complicated and hard-to-grasp at first seem to make much more sense a day or two later, even if I haven’t done any additional studying. For example, when I first read about simplicial homology, it was very confusing and frustrating. But the very next day, everything just clicked, and it all made sense. It is somewhat difficult to describe how this change happened, but nevertheless I felt very comfortable with the material after sleeping on it.
There are two possible explanations for this, and I have seen different mathematicians arguing for both. The first possibility is that you just get used to the material. As the famous saying goes, “You don’t ever learn mathematics, you just get used to it.” While that quote seemed somewhat crazy to me 2 years ago, now I know why such an aphorism exists. There really is a process that most math students go through, where even very complicated ideas suddenly become very comfortable after a day or two. Proponents of this theory would hold that, where as upon first learning one is concerned with the reasons and justifications for the new ideas and theorems, one begins to shift focus on the results themselves rather than the reasons for them as comfort settles in. And by shifting focus to the “what” rather than the “why”, the topic seems “easier”.
The second explanation is that your subconscious works out details while you are sleeping, hence you have really understood the material better the next day, even if you haven’t deliberately studied. Apparently Poincare was a big proponent of this idea, and hence he never worked on problems for more than short spurts at a time. I think I have also read somewhere that Einstein was sleeping almost 14 hours a day when he was working on Relativity.
I think both explanations are plausible, and a little of both are probably true. One thing is for sure: cramming for math doesn’t work. And studying for extended periods of time also doesn’t work. Frequent breaks are needed to solidify the material, which is exactly what I am doing right now. 
The orientability of a manifold is a mathematical notion that is seemingly abstract and far-removed from reality. It is an issue dealt with in Topology, a modern branch of mathematics that studies properties of space. But in fact, it is one of the reasons why you and I could exist as living beings.
You see, the biochemical molecules that are responsible for sustaining life, amino acids for example, are chiral molecules. This means they have handed-ness. Chemists refer to them as enantiomers. For example, L-Leucine is different from D-Leucine in that they are mirror images of each other. And if you tried to take L-Leucine and rotate it around in space until it looked like D-Leucine, you couldn’t do it. They are mirror-images. And the difference is profound. Certain pharmaceutical drugs cure diseases in one enatiomeric form. Its mirror image could kill you. Indeed, that was exactly the reason why Thalidomide, which was originally designed to cure morning sickness, led to babies being born with no legs or arms. They used the wrong mirror-image form of the drug.
What does this have to do with the orientabiity of manifolds? Well, the mere fact that mirror images of things exist in nature that are not identical (have different chemical effects, for that matter) means that space is orientable. If space were a non-orientable manifold, this distinction would not exist.
Now, we are not actually certain if space is orientable or not. If it was not orientable, an astronaut could travel far into space and come back to earth to find that he is the mirror image of what he used to be! He would find his Heart on the opposite side of his body, and that mole on his left side of his chin would now be on the right! And if L-Leucine had traveled the same path that the astronaut had, it would have turned into D-Leucine! But not one know if such a thing is possible, hence we do not know if space is orientable or non-orientable. But, at least such a phenomenon does not occur nearby earth. Otherwise, we would have to fear for our lives to accidentally travel an orientation-reversing path somewhere in space!
According to a strange mathematical law, about 1/3 of house numbers have 1 as their first digit. The same holds true for many other areas that have almost nothing in common: the Dow Jones index history, size of files stored on a PC, the length of the world’s rivers, the numbers in newspapers’ front page headlines, and many more.
-http://www.physorg.com/news98015219.html
What an interesting discovery. According to the article, scientists have been unable to find any a priori reason why one should occur most frequently as the first digit in various numbers. Somewhat expectedly, this law does not hold for numbers that are randomized or distributed in some deliberate way such as lottery numbers or telephone numbers. On the other hand, numbers that are not necessarily randomized or distributed in a certain way, such as the surface area of countries, mostly start with one.
The coolest part about this story is how the law was first discovered: apparently the physicist Benford was looking through logarithm tables and found that the pages that have numbers starting with one was the most crimpled and used. This suggested to him that perhaps data starting with one ocurred the most frequently in scientific research.
Although nobody may understand why this law holds, it has been put to good use. For example, it is used to detect accounting forgery and election fraud. If Benford’s law is shown not to hold in these data sets, one may suspect foul play. Of course, this does not constitute direct evidence of data manipulation. But given that Benford’s law has held true in so many different cases, lack of such a distribution of number is at least grounds for some suspicion.
I wonder if this law has anything to do with the fact that we use base 10 in numbering things. Perhaps other numbers can be shown to occur more frequently in the first digit if another base was used. If a similar result did not occur in other base systems, than this is evidence that Benford’s Law is solely an artifact of using base 10. On the other hand, if some form of this law still held in other base systems, then the mystery is still not solved. It makes sense that such a law would hold for house numbers, for example, only because you need the lower numbers to occur first before you get to the higher ones. But this makes no sense for things like area of countries.
If you are familiar with academic/scientific colloquiums and presentations, you will find this video absolutely hilarious. If you are not, well, then you are probably saner than any of the people laughing uncontrollably in this video.
I recently discovered a wonderful website: http://www.metamath.org
The website is a project to derive all of math from set theory (The ZFC axioms to be more precise). Of course, formalists have been espousing the view that all of math is essentially a game of logic since the beginning. While this view is held “in principle”, an attempt to actually carry through with this seemingly ginormous task has been largely ignored since Russell’s Principia Mathematica. Of course the Principia is a bit outdated, and doesn’t explicity use the ZFC axioms. Hence, this web project is a long-needed one in my opinion. In addition, the database uses symbolic logic exclusively, hence the rigor of the approach is never in question. Plus, this allows anyone who has looked at the ZFC axioms to understand the proofs (actually parsing out what the proof and the theorem mean is entirely another matter).
Another cool thing about this project is that a computer program has been written that checks the proofs automatically! Of course automated theorem proving is something that has divided mathematicians. Some claim it removes the beauty and the “Art” of mathematics while others see it as the only way to do truly rigorous math.
A fun thing to do is to just browse the various theorems that are in the site’s archives. Some of them are extremely interesting. For example, there is a proof for the principal of equality (that x=x). While many think this is an axiom (the axiom of equality), it can actually be derived from the ZFC axioms, which do not include the equality principle. I think this is completely mind-blowing. Something as seemingly trivial as equality can actually be proven rigorously!
If I had more time, I might want to work on becoming conversant in symbolic logic so I can actually read this stuff.
In physics and intro calculus courses, the symbol dx is taught to be a literal infitesimal value. Hence, the derivative, dy/dx is considered literally to be a ratio of infitessimal values. The integral, with a dx at the end, is thought to be highly analagous to the summation, with the infitesimal term simply serving as another factor in the “infinite summation”. This sort of thing can be taken to the extreme: to calculate the derivative of a function, simply plug in x+dx into f(x) and do some algebra, keeping in mind that dx^2=0 but dx itself is not zero! Surprisingly, that gives you the right derivative! Of course all this is not rigorous at all. Doing algebra with infitessimals is problematic on many levels. There are many problems both mathematically and philosophically, not the least of which is the law of excluded middle.
But the fact that infitessimal algebra “works” suggests that it should be possible to make it rigorous. And indeed, this has been done by many mathemticians. One example is called Non-Standard Analysis, where the Real Numbers are extended to the Hyper-Real Numbers, where infitessimal values are added to the Real Numbers. Apparently, you need to deny the law of excluded to make this work though. Many mathematicians are also somewhat skeptical of the topic. Some have critisized that the theory suffers from ontological problems.
What is more interesting is that when Newton and/or Leibniz first invented calculus, they did so mostly using an algebra of infitessimals. Hence, their invention was not so rigorous or well-defined. It took work by many other mathematicians years later to do the “cleaning work”. Now, it is possible to make sense of infitessimal algebra without a rigorous theory if you make some ad-hoc arguments, and I suppose this is why it continues to be used in classes. I don’t think I have a problem with this, as long as some of the flaws are pointed out as well. If some mindless high school teacher is teaching calculus without all the philosophical implications of infitessimal algebra, it’s a shame.
One of the more interesting thing I have been pondering about recently is the distinction between logic and ontology in math. Mathematicians spend a good deal of time developing the theory (or the structure, if you will) of mathematics, but spend far less time worrying about ontology. The theory, of course, is developed using the rules of logic. It can be even said that all of mathematics is simply a subset of set theory, which is a subset of logic (formalists think that anyways). But ontology is more subtle because it is not certain whether you can prove a physical existence. The traditional empricist approach of experimentation does not work here because the “Existence” mathematicians speak of is more abstract than a mere materialistic existence. For example, do the Natural Numbers exist? Most mathematicians simply chose to postulate that such a set exists. However, it is unclear whether this can be shown by experimentation. There certainly is no clear way of showing it through logic. Hence, ontology seems to escape both empiricism and logicism. Should mathematics be even concerned with it?
I would argue whole-heartedly that mathematicians should be concerned with it. Because if ontology is not established, than the whole affair of mathematics is useless. For example, I recently read something by the famous mathematician Alain Connes that there is a whole branch of non-standard analysis that is logically sound and very beautiful until you realize that none of the things they are talking about actually exist. So it is a useless affair. (Or is it?) Clearly, ontology is important in this case. But the trouble seems to be that there is no satisfying way to investigating the issue.
Hi everyone. I guess I’m in the blogosphere now. I read somewhere that over a million blogs are created everyday. So I guess this is one of them. I hope to share my thoughts and frustrations on this blog in the future.
So I will begin by introducing myself. I am currently a sophomore at Swarthmore College. I recently declared a major in Mathematical Physics. In addition, I am planning on a music minor. My interests in math include Foundations, Axiomatic Set Theory, Point-Set Topology, and Geometry (Differential). My interests in physics include General Relativity, Newtonian Mechanics, Maxwell’s Eqations, and philosophy. In terms of music, my favorite genres are the Baroque and Progressive Metal.
I recently discovered a great artist, called Neal Morse. He is a prog-rock composer, and his concept album “?” (yes, its called question mark) is one of the greatest albums I have heard in a long time. It’s not prog-rock per se, but it uses alot of mannerisms of that genre without the heavy guitar riffs. I have been listening to it constantly. It’s quite catchy, but it a good way. Now it is a Chrisitian album, which turned me off from it a little bit. Some parts do get preachy. But then I remind myself that most of Bach’s wonderful choral music are also religious in theme!
