Huperborea

On his blog, Jerry Coyne, who is supposed to be a famous evolutionary biologist, wrote the following in response to an article by Karl Gibberson titled Mathematics and the Religious Impulse:

Mathematics is, of course, a logical system invented by humans, and so has to “work”. One could equally well ask, “Why does logic work?” But if Giberson is asking, “Why does math help us understand the world?”, that seems equivalent to asking “Why does nature obey laws?” One answer is that if it didn’t, we wouldn’t be here to ask the question. But maybe I’m missing something. Yet consider this: if nature didn‘t obey laws, would we see that as evidence for no God? Of course not! In fact, the temporary and local suspension of physical law is precisely what a miracle consists of, and miracles, of course, are evidence for God. So when physical laws are obeyed, God’s working, and when they’re broken, God’s working too. Perhaps there’s some intermediate degree of lawlessness that would convince the faithful that there is no God?

My response:

Professor Coyne,

I have no doubt that you are a giant among fruit fly ejaculate researchers but you are an ignoramus concerning mathematics (and other subjects, from what I’ve observed.) Humans did not “invent” mathematics. Not even Euler, Cauchy, Gauss, or the other greats of mathematics could have “invented” the fabulous results of complex analysis. And abstract algebra was advanced without any application in mind but it turned out to be crucial for physics.

But don’t let any of that stop you from babbling about subjects you know nothing about or spinning just so yarns.

Labels: mathematics

Over at Pharyngula, a pseudointellectual with "just enough of learning to misquote," FrankT, attempted to criticize Gödel's Ontological Argument, a feat which is manifestly above his intellectual (I use the word loosely in reference to him) pay grade. I have copied the cognitively-impaired poser's comments and interspersed my responses below.

There are five axioms to it, and I will bold the ones that are bullshit:

1. A quality can be uniquely positive or negative.
(fails even a cursory Buddhist or Aristotelian analysis, but moving on)

This is a textbook example of pseudoargumentation. Moreover, it is positive or non-positive. Apparently, FrankT is unaware of the concept of zero.

2. If a quality is uniquely positive, then the qualities implied by having that quality are positive.
(I think this one slides, because it's basically definitional, not that it matters because I cannot name a single uniquely positive trait)

FrankT's inability to "name a single uniquely positive trait" is not my problem, nor is it Kurt Gödel's.

3. If two qualities are positive, both qualities together are positive.
(while I like "delicious" things and "intelligent" things, I would prefer that intelligent things not be delicious)

This is just as vapid a response as the tea "counterexample" offered by a guy at the Sierra Q&A. To quote Christopher Small:

In ordinary language, we might be inclined to say that one thing is greater than another if
the former has some positive attribute that the latter lacks. We might disagree with each
other as to the ranking of things or objects according to their value, but we must inevitably
make such judgements, whether we regard them as objective or not. In view of the
ambiguity of such concepts, it is important to understand what Gödel meant by a positive
attribute. In his own words he said that the operator Pos could be interpreted in a
moral-aesthetic sense, or in the sense of pure attribution. The concept of a predicate
being positive in a moral or aesthetic sense, provides no difficulty, at least initially.
Clearly, if Fx means that x is beautiful, we would be willing to accept that F is positive
in an aesthetic sense, even if we disagree in our judgements about beauty. If Fx means
that x is virtuous, we might grant the same, even if we have no idea what virtue is. But
what is meant by “pure attribution?” By “pure attribution,” Gödel states that we are to
understand that a predicate attributes some quality to an individual, and that the quality
contains no element of “privation.”

"Deliciousness" is not a positive property in either sense. Thus, FrankT's "counterexample" falls apart.

4. All properties are uniquely positive or negative and not both.
(this one is just actually laughable, so whatever)

Yet another textbook example of pseudoargumentation from intellectually-bankrupt FrankT. Moreover, as I noted above, it is positive or non-positive. Apparently, FrankT is unaware of the concept of zero.

5. Existence is a quality that is positive.
(existing would seem to have an overall degree of positiveness equal to the overall positiveness of the thing that either did or did not exist. Something baleful like a werewolf would be more negative if it existed than if it did not exist, while something awesome like the invisible pink unicorn would be more positive if it existed. And of course, something completely inconsequential like the celestial teapot would be just as meaningless if it existed as if it didn't)

Necessary existence is a positive, you damn moron. Gödel's Ontological Argument employs modal logic.

So basically it's really easy to refute, because all of the premises are wrong. Furthermore, there is a core problem with the logic, which is that just because an infinite series summation of positive traits would, given those premises, sum to the existence of an all powerful all whatever god-thing, doesn't mean that such an infinite series ever actually starts.

Achilles catches the tortoise, because Zeno's paradox is answered by Leibnitz and Newton and the infinite series of half distances does complete and total up to a whole. But Achilles does not catch every tortoise, he just catches the ones he chases. He has to start a series before he can finish it. And there is no reason to believe that anything ever started amassing an infinite series of all possible positive traits, even if there was such a thing.

Now, the Ontological Argument goes that basically by imagining the collection of all possible positive traits that you have in fact started that process - that your very own hubris is Achilles and that when you catch the tortoise it will be an omnipotent god. That is a level of solipsism and hubris that defies ready comprehension. But you can see why it might have appealed to St. Anselm, Leibnitz, and Kurt Gödel. Because they were all whack jobs who spent all their time thinking about infinity.

...

The fact that the Ontological argument is a total failure has little to do with the fact that it uses modalities. It's that both the premises and the logical chain after it are wrong. Perfectly valid attacks against it include attacking virtually any of the premises. But also the fact that infinite series summation just doesn't work that way. Convergent series reach conclusion and give real answers. But divergent ones don't. In the real universe there are no infinitely large rocks, infinitely large amounts of energy, infinitely large spaces, or infinitely powerful forces. Everything, even amongst the really stupidly large things, is finite.

So if someone has a series that includes infinite power or infinite anything else, then that series by definition never completes. So since it never finishes acquiring all the traits, there's no reason to believe it ever acquires "existence" either.

This is the most moronic part of an exceedingly moronic screed. So moronic, in fact, that it is not even wrong. (Yet, FrankT was still able to hoodwink a number of dupes over at Pharyngula, which speaks volumes.) First of all, Anselm, Leibniz, and Gödel were great thinkers, not "whack jobs," and it is offensive to see them labeled as such by FrankT, a noxious mediocrity. Moreover, an "infinite series summation" doesn't enter into it. Axiom 3 (as listed by FrankT, which follows the presentation on wikipedia) simply states that if p1, p2, ..., pn are positive then the collection, i.e.,
p1 AND p2...AND pn is positive. There is no infinite sum here.

Now, if we were talking about probability (we're not) and we looked at the probability of p1 AND p2 ... AND pn, i.e., the intersection (for the sake of argument treating the pk's as events, which they aren't), then we would have an infinite product of probabilities as n approached infinity provided the pk's were independent (again, treating the pk's as events, which is not the case). Even that stretch does not give an infinite sum, though. To arrive at an infinite sum, you would have to look at the probability of p1 OR p2...OR pn, i.e., the union of the pk's, which would be an infinite sum of probabilities as n approached infinity provided the pk's were mutually exclusive.

However, we are not dealing with probability and even if we were, the closest analog is an infinite product, not an infinite sum. (Incidentally, both the infinite product and the infinite sum I mentioned would fall in the interval [0, 1]). How this brain-dead pharyngulite came to associate Gödel's Ontological Argument with an "infinite series summation" is beyond me.

Incidentally, for those readers who do not know the difference between independent and mutually exclusive events, I recommend the following video I posted to youtube a couple of years ago.

And while Kurt Gödel ended his life of starvation because he was mortally afraid of being poisoned and would not eat unless his wife prechewed his food to assure him that it had not been poisoned, he was very good at thinking about infinity when he wasn't waiting for his wife to get out of the hospital so that he could eat again.

...

Which is why I was so surprised that our creationist friend stood up and said that his best argument for a god was Gödel's ontological proof. Because that one is a set of magic words written by a crazy man after he started wasting away into madness. That's like saying that your best argument for something is the last mad scratchings of Dr. Herbert West.

This is a low-blow but one I would associate with a mind of FrankT's caliber. Not only that, but it competes with his "infinite series" nonsense on the scale of utter stupidity. Kurt Gödel formulated his ontological argument long before he started suffering from paranoia and even if that were not the case, there is no reason to believe it would have adversely affected his mathematical work. The fact of the matter is that Gödel was far, far more intelligent and competent in his final days than FrankT could ever hope to be on his best day.

Labels: God, mathematics, philosophy, probability, theism, theology

Okay, the following is nothing special (upper division undergrad stuff) but it is instructive:

The supremum of a nonempty set is an upper bound (not necessarily contained in the set) such that every element of the set is less than or equal to it and among all upper bounds of the set it is the least. For example, the supremum of the interval [1,5) is 5.


Now, say a set A is contained in a set B (i.e., A is a subset of B). That means if a is an element of A then a is an element of B. By definition, a is less than or equal to the supremum of A. Also, since a is an element of B, then a is less than or equal to the supremum of B. That means the supremum of B is an upper bound for the set A. (The element a is arbitrary.) However, the supremum of A is the least among the ubber bounds, so it must be that the supremum of A is less than or equal to the supremum of B.

Quod erat demonstrandum

Incidentally, another, more descriptive, name for supremum is least upper bound.

Labels: mathematics

Moses Maimonides is among the greatest of Jewish thinkers.

(M. FRIEDLÄNDER translation)

Book III, Chapter LI

My son, so long as you are engaged in studying the Mathematical Sciences and Logic, you belong to those who go round about the palace in search of the gate. Thus our Sages figuratively use the phrase: “Ben-zoma is still outside.” When you understand Physics, you have entered the hall; and when, after completing the study of Natural Philosophy, you master Metaphysics, you have entered the innermost court, and are with the king in the same palace. You have attained the degree of the wise men, who include men of different grades of perfection. There are some who direct all their mind toward the attainment of perfection in Metaphysics, devote themselves entirely to God, exclude from their thought every other thing, and employ all their intellectual faculties in the study of the Universe, in order to derive therefrom a proof for the existence of God, and to learn in every possible way how God rules all things; they form the class of those who have entered the palace, namely, the class of prophets.

Labels: God, mathematics, philosophy, theology

I admire Kurt Gödel's Ontological Proof of God. (Just as I admire Gödel himself.) Dr. Christopher Small has a good discussion of Gödel's proof here.

Labels: God, mathematics

I recommend this excellent video about elementary mathematics education presented by M.J. McDermott. In the video she addresses some of the many problems with mathematics education in this country. (Although, she is specifically talking about Washington's mathematics curriculum these problems are not endemic to Washington.)

(via Telic Thoughts)

Labels: mathematics

John Lynch made me aware that today is the sixty-fifth anniversary of the tragic death of mathematician Felix Hausdorff. As I commented on his blog, I wish someone would have found Hausdorff a position elsewhere so that he did not feel compelled to remain in Germany.

Labels: mathematics

Quote: mathematical analysis is a language for learning the internal harmony of the world, the only true objective reality

Quote: mathematical analysis is primarily the study of temporal and spatial frames

Quote: objective reality is that which is common to many thinking beings; it can only be the harmony expressed by mathematical laws [»poinH_1905, OK]

Quote: mathematics is an objective, ideal reality that is neither subjective nor physical [»daviPJ_1981]

Quote: mathematics is independent of the existence of material objects; mathematical existence means free from contradiction [»poinH_1908, OK]

Quote: mathematical concepts have absolute truth and a Platonic existence; e.g., the Mandelbrot set [»penrR_1989]

Quote: arithmetic laws are the laws of the laws of nature

Quote: although the senses can be confused, corporeal things contain what is clearly understand; at least as the objects of pure mathematics [»descR_1641] Source I am a Platonic Idealist, and I see Mathematics as a triumph of Idealism.

Labels: mathematics, philosophy

Least concave majorant--The smallest concave function lying above a function of interest. I learned about the concept today in my stochastic differential equations class, specifically in the context of optimal stopping times.

Labels: mathematics

One of my (favorite) instructors compared Lebesgue integration to Riemann integration with the following analogy:

Say you dump your change jar on the floor. Riemann goes along and adds up each coin as he picks it up; whereas Lebesgue counts the number of pennies, nickels, dimes, quarters, etc. and multiplies each total by its corresponding value.

Labels: mathematics

Roger Bacon is among my favorite Medieval scholars. I especially like what he wrote concerning the importance of mathematics:

In the mathematics I can report no deficience, except that it be that men do not sufficiently understand the excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures; so in the mathematics, that use which is collateral and intervenient is no less worthy than that which is principal and intended.
Quoted in J Fauvel and J Gray, A History of Mathematics: A Reader, 1987.

Et harum scientiarum porta et clavis est Mathematica.
Mathematics is the door and key to the sciences
Opus Majus

For the things of this world cannot be made known without a knowledge of mathematics. For this is an assured fact in regard to celestial things, since two important sciences of mathematics treat of them, namely theoretical astrology and practical astrology. The first ... gives us definite information as to the number of the heavens and of the stars, whose size can be comprehended by means of instruments, and the shapes of all and their magnitudes and distances from the earth, and the thicknesses and number, and greatness and smallness, ... It likewise treats of the size and shape of the habitable earth ... All this information is secured by means of instruments suitable for these purposes, and by tables and by canons .. For everything works through innate forces shown by lines, angles and figures.
Opus Majus

Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of the world.
Quoted in C B Boyer, A History of Mathematics (New York 1968)

There are four great sciences ... Of these sciences the gate and key is mathematics, which the saints discovered at the beginning of the world.
Opus Magus

... mathematics is absolutely necessary and useful to the other sciences.
Opus Magus

Neglect of mathematics works injury to all knowledge, since one who is ignorant of it cannot know the other sciences of the things of this world. And what is worst, those who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy.

Source

Labels: mathematics, philosophy

(These are from the lectures of my undergrad mentor, Dr. Meyer.)

Terms from Statistics

1. Variable-a characteristic of an object or individual.

(a) Political affiliation of a registered voter.
(b) Weight of a certain type of package ready to be shipped.

2. Experimental Unit- an object or individual on which a variable is observed. (Observations are called measurements or data.)

(a) Each registered voter
(b) Each such package

3. Population-the group or collection of all possible observations.

(a) The collection of political affiliations of every registered voter.
(b) All weights that are conceivably possible for such packages.

4. Sample-subgroup of the population.

(a) Political affiliations of a selected group of registered voters.
(b) Weights of a selected group of such packages.

5. Univariate data-result of measuring one variable on a single experimental unit.

(a) Political affiliation of a single voter.
(b) Weight of one single such package.

6. Bivariate data-result from observing two variables on a single experimental unit.

(a) Political affiliation and gender of a registered voter.
(b) Weight and volume of a certain type of package ready to be shipped.

7. Multivariate data-results from observing two or more variables on a single experimental unit.

(a) Political affiliation, gender, and age of a registered voter.
(b) Weight, length, width, and height of a certain type of package.

Terms from Probability

1. Experiment-an unambiguous, repeatable process by which an observation (measurement, datum) is obtained.

(a) Process of selecting a registered voter.
(b) Process of selecting a package.

2. Sample space-set of all possible distinct observations.

For the roll of a six-sided die, S = {1,2,3,4,5,6}

Labels: mathematics, statistics