tRNA message-unit Magic Code?

It is unclear to me whether this is an argument for or against interdisciplinary education.

Should You Be Attacked by Marauding Visigoths

Hopefully the first in a series of posts on the subject, act two of last week's This American Life (now available as an mp3) has some important advice about defensive weapons systems you might wish to install. It's the bit after the story about Peter Pan.

Two-Hour Topology Talk

Pardon the alliteration. I spent the second half of last week preparing for my talk on Friday. I gave a two-hour introduction to General Topology. I don't think it went that badly, except for a few moments where I blew a proof (actually, I just left out a quantifier that the proof hinged on. Kelley did the same, but I clearly wasn't quite thinking straight when I copied the proof out to present), and a couple definitions.

Speaking of definitions, it's interesting that one can give an engaging talk for quite a long time only covering basic definitions without providing any real results. This was the second talk I've given for PL Jr; and the second for which I've done precisely that. The last one was on Universal Algebra. I've heard it said before that there are 3 kinds of mathematicians: Algebraists, Topologists, and Analysts. I'll be very surprised if I hit the trifecta.

I think the best line of the day was when someone was trying to claim that the intersection of two open intervals could contain a single point, and hence be closed. I responded by pointing out that no matter how small the intersection was, it was either empty or contained uncountably many points. "And," I contended, "I contend that a set with uncountably many points is not a singleton."

British Syntax

This from A History of Mathematics by Carl B. Boyer. My copy was plundered from John Wiley during their move to Hoboken.

The British contributors to algebra belonging to the generation of Abel and Galois, on the other hand, set out to establish algebra as a "demonstrative science." These men were strongly affected by the fact that England's analytic contributions lagged behind those of the Continent. This was attributed to the superiority of "symbolic reasoning," or, more specifically, of the Leibnizian dy/dx notation over the fluxional dots still prevalent in England.

I find this to be rather funny, but also a bit of an object lesson. Essentially, the British blamed their analytic inferiority on bad notation. It wasn't that they lacked the machinery to do analysis, just that their representation of it was difficult to work with to the point of impeding their progress. To put it another way, they blamed their lack of achievement on bad syntax. It was beaten into me in several math classes in college that good notation leads to insight. I don't see why a programming language's syntax should be thought of any differently.

Observations

It occured to me recently that there are no power sets of cardinality ℵ₀.

A question the answer to which did not occur to me recently: does there exists an infinite cardinal n such that there exist more (or less) than 2ⁿ functions from n → n?

Kleene the Platonist?

More from Kleene. I don't know that I've read enough mathematical epistemology, nor do I actually know that much about Kleene, but I find it interesting that he takes this digression in a book about formal systems to plead for meaning.

But in the case of arithmetic and analysis, theories culminating in set theory, mathematicians prior to the current epoch of criticism generally supposed that they were dealing with systems of objects, set up genetically, by definitions purporting to establish their structure completely. The theorems were thought of as expressing truths about these systems, rather than as propositions applying hypothetically to whatever systems of objects (if any) satisfy the axioms. But then how could contradictions have arisen in these subjects, unless there is some defect in the logic, some error in the methods of constructing and reasoning about mathematical objects, which we had hitherto trusted?

To say that now these subjects should instead be established on an axiomatic basis does not of itself dispose of the problem. After axiomatization, there must still be some level at which we have truth and falsity. If the axiomatics is informal, the axioms must be true. If the axiomatics is formal, at least we must believe that the theorems do follow from the axioms; and also there must be some relationship between these results and some actuality outside the axiomatic theory, if the mathematicians' activity is not to reduce to nonsense. The formally axiomatized propositions of mathematics cannot constitute the whole of mathematics; there must also be an intuitively understood mathematics. If we must give up our former belief that it comprises all of arithmetic, analysis and set theory, we shall not be wholly satisfied unless we learn wherein that belief was mistaken, and where now instead to draw a line of separation.

This Post is False

Via Kleene's Introduction to Metamathematics, a nice restatement of the Russell Paradox, apparently due to Russell:

A property is called 'predicable' if it applies to itself, 'impredicable' if it does not apply to itself. For example, the property 'abstract' is abstract, and hence predicable. the property 'concrete' is also abstract, and hence is impredicable. What about the property 'impredicable'?

Quine Quotes

I picked up a copy of Set Theory and Its Logic by one Willard Van Orman Quine. Few good quotes thus far (emphasis mine):

If the sentence represented by 'F x' happens to contain a quantifier capable of binding the y of 'F y', then an evasive relettering of the quantification is to be understood...

I think evasive relettering is a much more user-friendly term than α-conversion.

From the preface to the First Edition (emphasis still mine):

Because the axiomatic systems of set theory in the literature are largely uincompatiblewith one another and no one of them clearly deserves to be singled out as standard, it seems prudent to teach a panorama of alternatives. This can encourage research that may some day issue in a set theory that is clearly best. But the writer who would pursue this liberal policy has his problems. He cannot very well begin by offering the panoramic view, for the beginning reader will appreciate neither the material that the various systems are meant to organize nor the cconsiderationsthat could favor one system in any respect over another. Better to begin by orienting the reader with a preliminary informal survey of the subject matter. But here again there is trouble. If such a survey is to get beyond trivialities, it must resort to serious and sophisticated reasoning such as could quickly veer into the antinomies and so discredit itself if not shunted off them in one of two ways: by abandoning the informal approach in favor of the axiomatic after all, or just by slyly diverting the reader's attention from dangerous questions until the formal orientation is accomplished. The latter course calls for artistry of a kind that is distasteful to a science teacher, and anyway it is powerless with readers who hear about the antinomies from someone else. Once they have heard about them, they can no longer submit to the discipline of complex informal arguments in abstract set theory; for they can no longer tell which intuitive arguments count. It is not for nothing, after all, that set theorists resort to the axiomatic method. Intuition here is bankrupt, and to keep the reader innocent of this fact through half a book is a sorry business even when it can be done.