Archive for the ‘Mathematics’ Category

Induction from observable facts

November 15, 2010

Einstein once made a fascinating casual remark that relativity and quantum could never be discovered inductively from observed phenomena (this is from memory, so I may be wrong)… I got the impression that when Einstein did his “thought experiments” and riding on a beam of light, that he was somehow viewing the kaleidoscopic panorama of possible axiomatic systems like Riemann, fixed-point, etc. when suddenly, serendipitously one notices a coincidence a match (sort of like Maxwell changing his equations based upon aesthetics and suddenly stumbling upon something that works) …. I am saying this with the view that some mechanical heuristic formulaic approach to grind out solutions is a consummation devoutly to be wished.. but impractical. It is sort of like Godel or Wittgenstein pointing out a fundamental recursive weakness in linguist or symbolic axiomatic systems… I suppose I am reacting to the phrase “making conclusions from knowledge of facts” … and then Euler’s simple formula sparking 30 or more years of string theory research…’s_formula
… or Galois pouring out in one night before a fatal duel material which takes a graduate student 3 months to master… (so I am told)


Young Isaac Newtons Memory Feat

November 9, 2010

I had heard the story about the young Newton who memorized Euclid’s first theorem, then demonstrated it from memory, then memorized the second theorem, then from memory demonstrated the first AND second, proceeding in a like fashion (with each new theorem memorized he again demonstrated from the first to that current theorem.) I forced myself to do the same thing. At the end of the Freshman year I was able to stand at a blackboard in McDowell hall and demonstrate from memory the first 5 books of Euclid. When we came to Book V (and it was Spring and we were young and romantic) NO ONE was able to do the long theorems on ratios, I did one, and then when no one could do the second the tutor asked me to do the second. I sensed some hostility in the glances of my audience. Anyway, now I can not remember any of it. I can only remember what it FELT like to memorize all those theorems. When I set out to do this feat I thought it would be impossible. I was surprised that I actually succeeded.

Einstein on Intuition

October 7, 2010

Nyc [yes it is a first name pronounced like NICK]:

Here’s some more from Einstein on the subject:

“Man tries to make for himself in the fashion that suits him best a simplified and intelligible picture of the world. He then tries to some extent to substitute this cosmos of his for the world of experience, and thus to overcome it.

He makes this cosmos and its construction the pivot of his emotional life in order to find in this way the peace and serenity which he cannot find in the narrow whirlpool of personal experience.

The supreme task—is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction.

There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them.”

Not that I am at all clever about math or physics for I am not; but I once stumbled across an utterly startling remark allegedly made by Einstein to the effect that NO ONE could ever have arrived at Relativity through inductive reasoning from observed phenomena. I took this to mean that someone of the stature of an Einstein or Godel or Tesla or Euler simply browse through the kaleidoscope of axiomatic systems in their mind until they stumble upon a curious correlation between what is axiomatic and what is phenomenal. Imaginary number were initially so named because no one dreamed that there might be some analog in the real world but I am told that wave mechanics requires the imaginary square root of -1 [ i ]. An Annapolis tutor who never made tenure, William Pitt, told me about “model theory” which is sort of like what this is all about. Namely, Copernicus’s epicycles accurately predict the positions of heavenly bodies to within a certain degree of accuracy but is there an actual physical analog in nature to an epicycle?

Computer algorithms

July 11, 2010

There’s an amazing paper by what’s his name MacKenzie about the multiple definitions of “0” that computing has brought us to. Read it late last semester and was much impressed with his skill-set — for a SSK guy he really did his homework.

Here’s the abstract and citation: “The first part of this paper discusses floating-point arithmetic, as performed by
computers and advanced digital calculators. It shows that different computer arithmetics have been proposed, and that the nearest approximation to a consensual computer arithmetic – the Institute of Electrical and Electronics Engineers’ standard for floating-point arithmetic – had to be negotiated, ratherthan deduced from existing human arithmetic.
The second part of the paper discusses mathematical proof of the correctness ofprograms and hardware designs, which is increasingly being demanded for systems crucial to safety or security. It argues that this extension of the domain of application of mathematical proof involves the negotiation of what a proof consists in. In 1987, this argument led the author and colleagues to predict a legal case concerning the nature of mathematical proof. Such litigation took
place in 1991; the key point at issue is described. More general disagreement about proof as applied to computer systems is also discussed. A further prediction is made: that there will be litigation concerning not just what kind of argument counts as a proof, but over the internal structure of formal proofs.”
MacKenzie, Donald. “Negotiating Arithmetic, Constructing Proof: The Sociology of Mathematics and Information Technology.” _Social Studies of Science_ 23, no. 1 (Feb. 1993): 37-65.

William replies:

Barbara, thanks SO MUCH for taking the time to post this interesting abstract which I have read several times. I remember some time in the 1970s hearing about some space program failure due to a floating point inaccuracy in a compiler. For some reason you post has reminded me about and

I was also reminded of certain debates as to what constituted an admirable mathematical proof (the complaint that the 4-color proof was inelegant for using so much computer output or that Wiles’s 300 plus page proof of Fermat’s last theorem is simply too long.)

The litigation aspect is very interesting as well as disputes over intellectual property.

Morris Kine’s books is useful to inspire all these thoughts. I wonder if being/reality/universe is ultimately knowable in some reductionist sense in that it can be precisely digitized in some Grand Unifying Theory, or is the task of the mathematical-physician endless meaning that being/reality is something holistic which comprises more than all its seeming individual parts.

Furthermore, our insatiable desire to KNOW via dangerous experiments like C.E.R.N seems Faustian in that we are willing to risk destroying this world which we have in the hopes of completely understanding a world which we have destroyed through our process of analysis and industrial revolution.

When 9 children wearing green sneakers encounter one child wearing white sneakers, hatred, ad hominem, bullying and ostracism or even physical violence ensues. Each of our faculties has some natural purpose which promotes our survival. I believe it is the 4th Psalm which says something like “Be angry and sin not; weep upon your bed for what you say in your hearts” (from the Greek Septuagint version.) Ancient theologians point to this form of anger as righteous. But each of our faculties taken out of context and moderation and proportion becomes something very destructive; hunger becomes gluttony, natural desire becomes lust; curiosity and inventiveness become overweening pride.


Those who choose to take seriously the Judaeo-Christian scriptures cannot ignore the assertions therein that 1.) the poor shall always be with you and 2.) until the last days their shall always be wars and rumors of war. These two facts (if they are indeed facts) are puzzling in the face of injunctions to the virtue of peace-making (that they are blessed) and alms-giving (to provide comfort to the poor.) We know from the commandment “Thou shalt not steal” that private property and hence capitalism is scriptural and communism in its purest form is not (by the way the 900 page Catholic Catechism stresses this point about private property.) We see from the Book of Revelation that up until the very last there shall be merchants who bemoan the lack of sales, which implies that there is free-market capitalism until the end. AND YET, we notice that small infants seem to love and trust indiscriminately and we are told that UNLESS we become LIKE these small children we shall in no wise enter the kingdom of heaven (and kingdom implies that heaven is not a democracy.)


Kant was the very first to conjecture that some “stars” are actually entire galaxies of stars. Some suggest that the results of quantum and relativity prove Kant wrong in his assertions regarding space and time. I am intrigued by something that Sartre says somewhere to the effect that if it were not for NOTHINGNESS (and perhaps undecidability) then everything would be such a plenum that freedom would not be possible; hence we are DOOMED to be free. Now Talmudic thought takes a similar tact in the notion of Tsim-Tsum or “divine withdrawal;” namely, that G-d is such a fullness/completeness that G-d must CONTRACT or withdraw to make a MAKOM (place) for being/reality. This is why it is written that G-d is not IN the universe; rather G-d is the PLACE (Makom) of the universe.

Somehow undecidability and indefiniteness and uncertainty are at the heart of our existential freedom. The ball is ALWAYS in our court and the buck NEVER stops.

I once blogged about mortality and discord as survival advantages of our species. If we were all capable agreeing upon one single important issue then we might not have chosen to wander and diversify into every ecological and ideological niche. And the fact that individuals die (are mortal) means that there will always be idealistic youths whose kaleidoscopic and infinitely varied imaginations will always see the same old things in new and different ways.


I am glad. I spoke at some length with my Filipina Catholic wife about your concern for this Muslim woman and your anti-abortion stance as well as your conservative political views. I imagine you share much in common with her since she goes to Mass every day and prays the Hours of Divine Office (I may be inaccurate in this phrase). And I mention that she is from the Philippines because anyone who knows that culture knows that for the majority Roman Catholic piety is deeply ingrained in the very fabric of their character (just as Theravadin Buddhism is for a majority of Sri Lanka and Hinduism is for a majority of those in India.)

I realize that I am very much to the Left and perhaps not most logical choice for your FB friends list, but I would jump to the assistance of anyone with a need or question such as yours regardless of their political or ideological affiliation and I do feel it is most constructive for EACH of us to seek out the association of those who greatly differ from us in important regards rather than to surround ourselves only with those who are in like-minded agreement with us. It is only the tension of our diversity which sparks and inspires us out of the rut of our complacency.


Does mathematics deal with independently existing “Platonic” eidetic reality OR is mathematics merely the byproduct of human conceptions, constructions and methods of axiomatic predication?

Justine writes:

Do you mean “do numbers and their relationships exist independently of human consciousness?” Given the interaction of subatomic particles to make up elements, I would say yes, though it does not seem a very useful question. (Rather like does the tree falling in the woods make a sound?)

William responds:

Justine, I appreciate the spirit and good intention with which you post and the relativistic koan-like problems inherent in language, but the very human enterprise of model theory and the quest for a G.U.T. (Grand Unifying Theory) and string theory are very real and will ultimately result in a positive resolution or else a Godel-like demonstration that a such a resolution is impossible, but the likes of you, Justine, and I, William, … the likes of us are not of sufficient intellectual stature to realize such a resolution on our own steam. We shall only appreciate the truths which a select few have fully comprehended while watching a Nova episode or someone like Carl Sagan. BUT, Justine, I suspect that the intellectual character and stance of those who DO succeed in such a resolution will share more in common with MY essential nature than with YOUR essential nature since, with all due respect, you content yourself with dismissing the importance of some question which bothers you through a casuistic maneuver and then you remain right there resting upon your tinsel laurels, while my nature is to always ignore people like you and Gorgias and like the Energizer Bunny I just keep on going (trying.) Now I know what I have said will seem a bit harsh or unkind but it is my desire to address ideas and not individuals and I have no Dale Carnegie aspirations to “win friends and influence people.” But I do value your interest and readership. Thank you!
link and read the article carefully.

One of the most important questions raised in the above link is:

Is human understanding unlimited?

Anyone who feels this question is trivial or impossible to discuss is not really fit to enter into the discussion. Socrates put it in a nutshell when he said that those who presume to already know the answer or all the answers will not make a sincere effort at inquiry.

I went through 4 years of St. John’s (and I would do it all over). I met a few people who were truly intellectually honest and would directly and aggressively take a stand on something important. But I met a lot more phonies who made it through 4 years just wrinkling their nose in seminar and saying “but I dont GET it, how can you say….”

All of the world is basically divided into a few LIMB CRAWLERS and a multitude of LIMB SAWYERS.

A limb crawler has the courage and ambition to crawl way out on a limb, and TAKE A RISK, assembling together many and various points and passages in order to assert something new, something important.

LIMB SAWYERS are a parasitic species. They cannot exist without a LIMB CRAWLER. They hack away at whatever constructive effort the limb crawler is making. If they succeed, what they succeed in is simply refuting the limb crawler.


Jay writes:

I always felt that questions of existence were ill or vaguely posed. What would be the criteria to meet in order to “exist” in the eidetic reality?

Steven writes:
Ms. Saracen and Mr. Hales seem to have a point, and Mr. Buell (despite being spot-on)should seem more contrite as a result.

William replies:
personal friendship between Einstein and Kurt Godel. Search on articles which explain the fundamental difference in the way that Einstein saw mathematics vs the way Godel saw it. I read a lot of that a long time ago and it helped me to see some things. Godel definitely held the Platonic view that the objects of mathematics had some ideal existence (like Plato’s FORMS) and that somehow reality was a shadow or penumbra. Einstein had the opposite view. Einstein also made one off-hand remark which I find utterly intriguing. Einstein said that “it is impossible for anyone to inductively arrive at the theory of relativity from empirical observations” (I am paraphrasing from memory what I read many years ago.) This means that although the existence of rabbits in no way violates the principles of quantum physics, one cannot study rabbits and inductively intuit quantum. Poker obeys the laws of statistics and probability but one would not study statistics and probability to learn how to play poker. I may be totally wrong in how I took Einstein’s off-hand remark, but here is how I took it. The human imagination is a kalaidescope of non-being (horned rabbits and unicorns) which can churn out a countless number of different axiomatic systems. Suddenly one day someone, or perhaps several people simply STUMBLE upon an odd coincidence in mathematics which closely resemble some empirical observations. Next they get something which kind of seems to work, (like Ptolemy’s epicycles)… you can make a lot of accurate calculations (up to a point) but THEN the question becomes, DOES OUR MATHEMATICAL model reveal something about the underlying nature of the observed phenomena (Ptolemy called in “sozein ta phainomena” saving the observations). Ptolemy got close, Copernicus closer, Kepler closer still, Newton seemed to be right on the money, but then things break down on an inter-galactic or sub-atomic scenario, Einstein gets closer still… but…. well, I think you get my point (and if you DONT get my point by now, then you dont have the foundation to even read about this stuff.) I am not the person to put a bib on anyone and spoon-feed them because I am intellectually and academically limited… in fact I am an over-achiever since I do not really have the credentials or training to speak authoritatively on these things…. THE MISTAKE that many people, younger people, make is to think that the they can find a thread like mine and then say…. so what, I dont get it, show me…. THAT IS NOT WHAT YOU ARE SUPPOSED to be doing with your life.. YOU are supposed to be digging deeply into all thats available and YOU are supposed to be giving ME a hand and helping ME out of the darkness of Plato’s cave reality and let me step into the light of pure being… so if all you can do is wrinkle your nose, say I dont get it, and ask me a few questions, then you are wasting your life and time.. you should go do something more constructive (what I don’t know… so dont ask me)…. One of my posts this week has to do with a mathematician with some work on an equation who met a physicist and he saw that it was closely related to quantum energy states…. THAT is an example of what Einstein was talking about I suspect…


No Steven, read what I just wrote, get your ass in gear, stop being a limb-sawyer (we have enough in the G.O.P) and start being a limb crawler… cause listening to negative people like you is not going to get me anywhere


Steven: Negativity is relative. To agree with you makes me a drone, to disagree with you makes me a sloth? I reject both options. I believe in careful reading. I even re-read JayJay’s and Justine’s responses to ensure that they weren’t sawing limbs. My assent prompted me to write those words, not a desire to gainsay what you had uttered. At the risk of sounding dronish, I agree with all the rest that you are saying, by the way.


I dont care whether you agree with me or disagree with me. I want you to stand on your own two feet and say something ANYTHING that does not depend on what I have said and dissect my previous statements and play petty word-games like the pre-Socratic sophists. Don’t you GET it? Don’t any of you EVER look in the mirror and read everything you have ever written for the past 10 or 20 years and see how empty it is. Do you think Christopher Hitchens has made great contributions to knowledge. I do not. I dont give a CRAP about what any of you agree or disagree with. Say something on your own. Go out on a limb. Take a risk! Tell me how it really is without quoting something I have said back to me. Boggle my mind! My mind is fairly puny so that should not be too much of a challenge, but at least TRY .. at least make the effort. Don’t be a bunch of parasites.



No student should ever be required to defend his ignorance (some of us are assembled here to overcome that, and not because we like the sound of our own typing). )I know a lot of people with puny minds. You would not guess that they are independent because their intellects are no greater than mine. It must be that “puny” is a relative term as well, and that in order for this puny mind to grow, I need the assurance that you don’t truly think of me as a parasite. I knew what you meant, but was not given the heads-up that this is a timed test.
I understand your frustration. Some just don’t get it. But if a thing is knowable, then, I believe, a teacher can guide the way to knowing it only if he recognizes the potential barriers to learning.



Oh for pitys sake, dont rage at them William, its counterproductive.

Personally I believe mathamatics operate outside human conceptions despite being defined by them. Take imaginary numbers for example. What is the purpose of them? To describe something we cannot see, much less hope to understand. Yet someone had to think ‘what if’ before they could be discovered, defined and accepted.

Math as a whole follows rules wether we want it to or not, as any child sitting in a classroom looking to the black( *ahem* i meant, of course, green)board will tell you. It is a tool, a language, and a state of mind all at the same time.

Anyone who wishes to master it has my well wishes, and ultimately my respect.



Yes, I feel like “what do we mean by eidetic existence” is interesting and worth considering, although I have thought about it a bunch and not come up with anything very coherent or persuasive (I’m in a math program). I actually thought that the zen koan had a lot of relevance to this as it seems to raise the question of how important independence from the observer is with regards to granting existence to an object. There is perhaps some solid thought related to this observer/object existence question done in the particular case of existence of Schrodinger’s cat in 3+1 dimensional space etc. etc.



I was thinking about Schrodinger’s cat. Bertrand Russell and Alfred North Whitehead attempted to place all of mathematics on a logical foundation with “Principia Mathematica” but failed so I am told for reasons that are connected with Godel’s Indefiniteness theorem. Bertrand Russell personally knew Wittgenstein and openly admitted that he felt dwarfed by Wittgenstein’s towering intellect.

This proposition 7 comes to mind for obvious reasons:
Where (or of what) one cannot speak, one must pass over in silence.

I suspect that human understanding IS limited but I am no mathematician or physicist.

I do remember the first time I learned that Andrew Wiles solved the Fermat problem and quite literally tears of joy and wonder streamed down my face.

Regarding my earlier remark about Tsim Tsum you might one day enjoy reading “The Life of Pi” novel (by Yann Martel) which is built around the notion of “divine contraction” and, I suppose, Jain Anekantavada (multipointedness).


Oh yes, I should have mentioned my thought about how Kant wanted to PUT AN END to any future metaphysical speculation in his Prolegomena. Hegel dreamed of a time of Absolute Knowledge where history (time) would cease since there would be nothing new. I suspect both Kant and Hegel considered the human capacity for understanding to be unlimited.

Morris Kline’s book about mathematical uncertainty seems to suggest both the imperfection of mathematics and also that mathematicians will never be unemployed in their Sisyphean task (like smoothing the bumps out of an infinite carpet and they always pop up somewhere else.)


Theory 1: God produces the universe. God creates the universe so we understand why there is a Universe but why God, what is God, where did God come from.

Theory 2: There is no such thing as God or purpose, but there is all this universe stuff which just happened and furthermore just HAPPENED to produce conscious beings who puzzle about all this.

Theory 3: The universe PRODUCES God. Bear with me. Big bang, stars, planets cooling evolution, intelligence, science, the sentient beings gradually BECOME as God, creating life, creating other universes, black holes, finally destroying everything which becomes another big bang to start all over again.

Theory 4: Take theory 3 and have in happening in black holes tucked inside black holes tucked inside black holes. This is like a Buddhist dependent co-arising on a grand scale. The God thing is prior to being/non-being/space/time/eternity. Universe constantly begets God which constantly begets universe. IF one could speak of the EXISTENCE of God then God would become part of the causal matrix and be subject to Aristotle’s Anangke necessity. Hence God is prior to being and non-being and time. God cannot EXISTED in the universe, so God contracts, Tsim Tsum, and a place (makom) is created for the universe to happen, a place where God IS NOT. So on the quantum level, every sub atomic particle wavering/flickering/pulsating between being and non being is an entire cosmos. These countless “worlds without end” emanate a kind of morphic resonance which affects and synchronizes each other.

… (just saying…)


Berlin argued that Johann Georg Hamann was one of the first thinkers to conceive of human cognition as language – the articulation and use of symbols. Berlin saw Hamann as having recognised as the rationalist’s Cartesian fallacy the notion that there are “clear and distinct” ideas “which can be contemplated by a kind of inner eye”, without the use of language – a recognition greatly sharpened in the 20th century by Wittgenstein’s Private language argument.

If the idea of a private language is incoherent, then it would follow that all language is essentially public: that language is at its core a social phenomenon. This would have profound implications for other areas of philosophical study. For instance, if one cannot have a private language, it might not make any sense to talk of private sensations such as qualia; nor might it make sense to talk of a word as referring to a concept, where a concept is understood to be a private mental state.

In order to count as a private language in Wittgenstein’s sense, it must be in principle incapable of translation into an ordinary language [note: compare this to the Muslim concept of Qur’anic non-translatability] – if for example it were to describe those inner experiences supposed to be inaccessible to others.

The private language being considered is not simply a language in fact understood by one person, but a language that in principle can only be understood by one person. So the last speaker of a dying language would not be speaking a private language, since the language remains in principle learnable. A private language must be unlearnable and untranslatable, and yet it must appear that the speaker is able to make sense of it.

A recurrent theme in Wittgenstein’s work is that for some term or utterance to have a sense, it must be conceivable that it be doubted. For Wittgenstein, tautologies do not have sense, do not say anything, and so do not admit of doubt. But furthermore, if any other sort of utterance does not admit of doubt, it must be senseless.