It's dead Jim. (this blog that is)

Old Links

I cleaned up the blog a bit and removed old links from the sidebar. This is a backup.

    "the long and short of steering in computer games"
    Good Navmesh / Pathfinding
    Reinforcement Learning basics
    Reinforcement Learning: An Introduction
    RICHARD BELLMAN ON THE BIRTH OF DYNAMIC PROGRAMMING
    Paul Tozour on Pathfinding
    Modern Pathfinding Techniques
    AI - Stochastic Plan Optimization in Real-Time Strategy Games
    Lighting - The Light Field
    Lighting - Derivation of the Rendering Equation
    Lighting - Jaakko Lehtinen - Foundations of Precomputed Radiance Transfer
    Lighting - Radiometry and Light Transport
    Opt. PS3 - Branching
    Opt. Superscalar / Superpipeline CPUs
    Opt. PowerPC - Computer Arch. course
    Opt. XBox360 - At Least We Aren’t Doing That: Real Life Performance Pitfalls
    Optimization - c99 / restrict / aliasing
    Assembly - Write Great Code — Understanding the Machine, Volume I
    AI - Automatically-generated Convex Region Decomposition for Real-time Spatial Agent Navigation in Virtual Worlds
    AI - Dude, Where's My Warthog?
    AI - Weighted Majority Voting
    AI - Killzone’s AI: dynamic procedural combat tactics
    AI - Stochastic Plan Optimization in Real-Time Strategy Games

Platonic solids, it is extremely sad they were never mentioned in school, nor in university!

"... were a quantum leap in the development of ..., a leap above the details of computation to a real of powerful abstract concepts. The power of these abstract concepts - ... - lies in their ability to capture general features of computation, so that the existence of particular computations can be proved or disproved without attempting to carry them out" (Stillwell)

And the future of 'standard' analysis was forged.

"At the time of Brouwer’s death it appeared that your choices were:
(1) accept Brouwer’s theories, give up most of mathematics and give up talking to most mathematicians; or
(2) accept Church’s thesis, give up analysis and give up talking to most mathematicians; or
(3) reject constructive mathematics entirely.
This was not a difficult choice for most mathematicians;" (Constructivity, Computability, and the Continuum, Michael Beeson)

The one moment when you ask: 'Is the constructed real number continuum really necessary?'

(Of course, only considered as enlightenment after you have really studied the continuum).

Intuition 'mismatch'

'To the criticism that the intuition of the continuum in no way contains those logical principles on which we must rely for the exact definition of the concept “real number,” we respond that the conceptual world of mathematics is so foreign to what the intuitive continuum presents to us that the demand for coincidence between the two must be dismissed as absurd. Nevertheless, those abstract schemata which supply us with mathematics must also underlie the exact science of domains of objects in which continua play a role.' (Weyl)

And Brouwer grins.

Your basic 'naive set theory' exercises.

Lately, I have been seeing much of Mathematics as Philosophy, and this new attitude, although hard to describe, has been a major factor in yet another totally new level of understanding what I have studied in the past, am studying, and will be studying.

If you wish to see the philosophical roots of each and every set theory exercise you have ever been given, to the point that, while reading the philosophical paper, which contains not one formula, you keep going 'oh, this was in fact turned into an exercise' and 'this one' and 'that one' all while realizing the very painful thinness to which the whole idea was reduced before it was given to you, if you wish to read a whole chapter explaining the philosophy behind similarity, yes, the same one that is thinned down and taught in you linear algebra book, I suggest the not at all easy, but very rewarding read 'Introduction to Mathematical Philosophy by Bertrand Russell'.

I have read most of it, and now finally I see where the ideas of transfinite of Cantor that are the basis of standard analysis come from, and I see where I disagree with them, (with the help of Wittgenstein), and I can finally zoom out and see both perspectives, (me being on the non-standard, and finitist side), and I can now 'go along with' standard analysis some more steps, maybe even very large strides (which despite my previous many reading about the 'foundational problem' I was not yet ripe enough to do), knowing what I am doing, what is being fed to me, how to translate into the finitist perspective, and in general where the whole differences are leading. Finally, after days of sleepless reading, and dozens months of doubt, a bit of peace of mind.

"Finitists like to maintain a zero level of gobbledegook. Analysts are usually happy with one-level of gobbledegook and category-theorists are comfortable with any amount of gobbledegook." (http://mathoverflow.net/questions/102237/what-is-the-status-of-irrational-numbers-within-finitism-ultrafinitism by Steve L. Cowan)

"Even while considering the universe to be finite, one can do mathematics symbolically as a game with a system of rules. If the game doesn't have enough pieces we just add new pieces, with new properties or allowed moves as required. All that matters is that the enlarged system is compatible with the old system; that the smaller game is a subgame of the large one; that the smaller system can be embedded in the larger one.

When does something exist ? Well if there isn't something from amongst the objects under consideration that has the properties we want then we just create new symbols and define how they relate to the old ones.

If we were a pythagorean and the only numbers that exist are rational numbers, then we wouldn't call 2√ a number, but if we were also finitist symbolicists then we could embed any collection of numbers into a collection that contains not only numbers but also "splodges" which is what we're going to call 2√. It's important to always keep in mind that 2√ isn't a number - it's a splodge. In this new system of arithmetic we've invented we can add numbers to splodges to get new splodges like 1+2√. What a fun game. Let's add some more splodges. We're bored with algebraic splodges so let's add some non-algebraic splodges like the one in your question. Of course that expression is a bit cumbersome so we'll give it the shorthand symbol π instead.

Given a splodge x, it would make calculus easier if there were a splodge x+o that was nearer to x than any other splodge, however that isn't possible so we embed the splodges in a larger system called the hypersplodges that contains not only splodges but also vapors, and contains not only the concept nearer but also the concept "nearer". Vapors like x+o are "nearer" to x than any splodge could ever be, and when you're finished using them they evaporate leaving just a splodge.

We want a splodge that satisfies x2+1=0 however there isn't one, so we embed the splodges in a larger system called weirdums in which we've added a piece called i with the rule that i2=−1, and under the new system we can "add" splodges to weirdums to get new weirdums like 1+i.

In solving differential equations we'd like a function which is zero everywhere except at a single point but which has a non-zero area under the curve. There is no function that behaves like this so we'll go to a larger system that contains not only functions but also spikes which do have the desired property because the larger system contains a rule about spikes which says they do. Conveniently certain calculations involving spikes cancel out leaving just functions.

A finitist or ultrafinitist shouldn't recognise the concept of infinite sets therefore the only sets are finite-sets and since all sets are finite the adjective finite is superfluous therefore from this point onwards we just use the term "set". Some people want to consider sets that contain things they haven't put in there themselves - which of course can't be done because a set only contains the items we've put there. So we embed the system of sets in a larger system that includes not only sets but also dafties. In this daft system the rules are that a daftie can have an "affinity" for things whether those things have been previously mentioned or not. Dafties have an affinity for things in the same way that sets contain things. A compatible embedding of a system of sets in the daft system means that a set has an affinity for the items it contains when the set is considered as part of the daft system, therefore by daft reasoning one can say things not only about dafties but also about sets. To each set one can attach a number. You can't do this with dafties so we embed the numbers in a larger system containing sinners and attach a sinner to each daftie. Sometimes there is a need for something that looks like a daftie but has no sinner - such things are called messes. A mess can have an affinity for collections of dafties that no daftie could have an affinity for. This could go on, but you need gobbledegook theory. The set system has zero gobbledegook. The daft system is level-1 gobbledegook. The messy system is level-2 gobbledegook. Finitists like to maintain a zero level of gobbledegook. Analysts are usually happy with one-level of gobbledegook and category-theorists are comfortable with any amount of gobbledegook.

There are two ways to compatibally extend a system:
1) a conservative extension adds new items but doesn't say anything new about the old items that couldn't be said before;
2) a progressive extension does say new things about the old items but only about things that were previously undecidable
P.S. We can combine the vapors, splodges and linedups in a system called the messysplodges but they haven't been studied much because they're a bit messy."

(http://mathoverflow.net/questions/102237/what-is-the-status-of-irrational-numbers-within-finitism-ultrafinitism  by Steve L. Cowan)

I hereby coin the term 'Alien Induction' and its primary use: 'Proof by Alien Induction'.

"By their peculiarites you know them", even if they are eventually just relations.

Mathematical checkpoint.

Last week, I had reached the last section of the 420 book (Linear Algebra, Hefferon) we are studying together with Tom. Also I have just now reached page 93 (Section 11, Roots, Irrational Numbers) in Zakon's 'Analysis Basics' (Our approach is very patient, complete, sequential and meticulous), along with a great deal of less meticulous reading in other books, papers and 'opinions'.
I can now say, one of the original questions that this blog carries as it's title, namely the square root of two, has been almost demystified!

The next adventures have already been planned quite some time ago: https://sites.google.com/site/77neuronsprojectperelman/jad/theplan.

As with every checkpoint, a huge huge thanks goes to my wife and love Lena.

"The more elementary the question, the more likely the answer involves a lot of philosophizing and bluster." (A crank?)

"Aliens are the Greek Gods of modern times" (Jad)

Why was 'what is the square root of two really?' the right question to ask.

"Bolzano points out that Gauss’s first proof is lacking in rigor; he then gives in 1817 a “purely analytic proof of the theorem, that between two values which produce opposite signs, there exists at least one root of the equation” (Theorem III.3.5 below). In 1821, Cauchy establishes new requirements of rigor in his fa- mous “Cours d’Analyse”. The questions are the following:
– What is a derivative really? Answer: a limit.
– What is an integral really? Answer: a limit.
– What is an infinite series a1 + a2 + a3 + . . . really? Answer: a limit.
This leads to
– What is a limit? Answer: a number.
And, finally, the last question: – What is a number?
Weierstrass and his collaborators (Heine, Cantor), as well as Me ́ray, answer that question around 1870–1872. They also fill many gaps in Cauchy’s proofs by clarifying the notions of uniform convergence (see picture below), uniform continuity, the term by term integration of infinite series, and the term by term differentiation of infinite series." (Analysis by it's History, Hairer).

And if you like pictures, here is an indirectly related one (http://en.wikipedia.org/wiki/Algebraic_number#The_field_of_algebraic_numbers).

The 'coordinate concept of dimension', very laudable (think Aristotle), but also, surpassed since centuries.

The right comment.

Commenting 'And now in English' is never the useful thing to say, the right comment is 'And now from the beginning'.

Are you a 'rigor mortis' kind of guy? or maybe even a 'rigor post mortem' one?

"Certain generalities seem to have been drawn from this, namely that a concern for rigor comes at the end of a mathematical develop- ment, after the "creative ferment" has subsided, that rigor in fact means rigor mortis. Weierstrass himself provides a good counterexample to this generality, for all his work on the spectral theory of forms was motivated by a concern for rigor, a concern that was vital to his accomplishments.

Weierstrass was dissatisfied with the kind of algebraic proofs that were com- monplace in his time. These proofs proceeded by formal manipulation of the symbols involved, and no attention was given to the singular cases that could arise when the symbols were given actual values. One operated with symbols that were regarded as having "general" values, and hence such proofs were sometimes re- ferred to as treating the "general case", although it would be more appropriate to speak of the generic case. Generic reasoning had led Lagrange and Laplace to the incorrect conclusion that, in their problems, stability of the solutions to the system of linear differential equations required not only reality but the nonexistence of multiple roots. (Hence their problem had seemed all the more formidable !) In fact, Sturm who was the first to study the eigenvalue problem (1) proved among other things the "theorem" that the eigenvalues are not only real but distinct as well. His proof was of course generic, and he himself appears to have been uneasy about it; for at the end of his paper he confessed that some of his theorems might be subject to exceptions if the matrix coefficients are given specific values. Cauchy was much more careful to avoid what he called disparagingly "the generalities of algebra," but multiple roots also proved problematic for him. As he realized, his proof of the existence of an orthogonal substitution which diagonalizes the given quadratic form depended upon the nonexistence of multiple roots. He tried to brush away the cases not covered by his proof with a vague reference to an infinitesimal argument that was anything but satisfactory.

It was to clear up the muddle surrounding multiple roots by replacing generic arguments with truly general ones that Weierstrass was led to create his theory of elementary divisors. Here is a good example in which a concern for rigor proved productive rather than sterile. Another good example is to be found in the work of Frobenius, Weierstrass' student, as I shall shortly indicate.
"
(The Theory of Matrices in the 19th Century, Hawkins)

Oh matrix you special substitution.

"Thus by the mid-1850's the idea of treating linear substitutions as objects which can be treated algebraically much like ordinary numbers was not very novel" (The Theory of Matrices in the 19th Century, Hawkins)