“Metaphor helps a human being to breathe in this rarefied atmosphere of Gods.” – Y. Manin
“Somehow or other, for me mathematical research is a discovery, not an invention. I imagine for myself a great castle, or something like that, and you gradually start seeing its contours through the deep mist, and begin to investigate something. How you formulate what it is you’ve seen depends on your type of thinking and on the scale of what you have seen, and on the social circumstances around you, and so on.” – Y. Manin
“… truths are derived from a few self-evident propositions, by a chain of flawless reasonings; they are imposed not only on us, but on Nature itself. By them the Creator is fettered, as it were, and His choice is limited to a relatively small number of solutions. … they have asked if the savant is not the dupe of his own definitions, and if the world he thinks he has discovered is not simply the creation of his own caprice. Under these conditions science would retain its certainty, but would not attain its object, and would become powerless.” – H. Poincare
“We shall also see that there are several kinds of hypotheses; that some are verifiable, and when once confirmed by experiment become truths of great fertility; that others may be useful to us in fixing our ideas; and finally, that others are hypotheses only in appearance, and reduce to definitions or to conventions in disguise.” – H. Poincare
“Damn life-cribbing…” – Anon
In an attempt to grapple with the concept of wisdom (in its full sense: the quality of having experience, knowledge, and good judgment; the quality of being wise / the soundness of an action or decision with regard to the application of experience, knowledge, and good judgment. / the body of knowledge and principles that develops within a specified society or period), one could make a worse decision than turning to two brilliant and ranging minds: Henri Poincaré and Yuri Manin. These are the stars we guide by.
I did not grow up in a household that privileged science and math. My folks both have tremendous strengths, but participation or interest in science and the manipulation of symbols are not in the mix. To the extent that I have any clue whatsoever with anything that even remotely resembles the mathematical (to my knowledge an open question), it has been beaten into me (yes, including self-flagellation) by the powers that be. This beat-down has been a privilege and an honor and although unorthodox, and essentially beyond good and evil, it has demonstrated results. There are still questions, as far as I know, regarding continuity, frequency, scale and scope. By the Unbearable Lightness of Being Theorem there is no way to know whether I could have obtained the same results via a different method. The method and the madness having chosen their victim, rather than the converse. I imagine that there is a heterogeneous opinion out there on such things (likely with a larger variance than the opinion on the person tap(p)ed); this is both no big deal and small peanuts. I can only say thank you to everyone involved, and a special thanks to those who participate(d) with big hearts. (It is beyond my comprehension how this stuff ends up on television, nor, at present, do I find it all that amusing. Interesting: yes, bizarre: definitely and, ok, a little amusing, but in a twisted, like, that’s fucked up ‘eh? sorta way. Perhaps some new data will change my perspective? Only time will tell. I imagine many delight in the house of mirrors set up when our TV blares last month’s audio feed and I acknowledge it in print that it is disorienting and destabilizing at best. (The preceding thing that just happened when I typed that sentence has never happened (to my knowledge) in the history of humankind. (Oh wait, there was Ellen the other day. Adjusting weights…) A real life historical moment here on Effluvia. Experimental data confirmed! Themes echoing into the void. [EDIT: sum [sick] 20 seconds later… Be careful what you wish for (dream up?)… But always, be grateful for what you get. I vaguely remember an idea for a novel I had in which the main character was to fragment, beyond all recognition, into echos of echos of himself via “the communications infrastructure“]? Dreams, these days, are made of reality. A human animal, caught on tape, in its natural [sic] environment, kind of like watching those shows on the nature channel, no? Like shark week or one of those lil’ critters tossed between killer whales, only better, because it’s human.) There. Isn’t that nice? Such an experience has yet to be named, as far as I know, I thusly dub it: Life-cribbing (hint: as in to crib someone’s crib). This can be seen as an act of power electronics.)
I was dumm-struck [sic] by a constellation of ideas I uncovered in Manin’s essay Mathematics as Metaphor and even the richness of the introduction to Poincare’s Science and Hypothesis.
The discrepancy between observed and calculated values is thus not regarded as a falsification of the law, but as a correction that the law imposes on our observations. In this sense, there is always a necessary difference between facts and theories, and therefore a scientific theory is not directly falsifiable by the experience. For Poincaré, the aim of the science is to prediction. To accomplish this task, science makes use of generalizations that go beyond the experience. In fact, scientific theories are hypotheses. But every hypothesis has to be continually tested. And when it fails in an empirical test, it must be given up. According to Poincaré, a scientific hypothesis which was proved untenable can still be very useful. If a hypothesis does not pass an empirical test, then this fact means that we have neglected some important and meaningful element; thus the hypothesis gives us the opportunity to discover the existence of an unforeseen aspect of reality.
… truths are derived from a few self-evident propositions, by a chain of flawless reasonings; they are imposed not only on us, but on Nature itself. By them the Creator is fettered, as it were, and His choice is limited to a relatively small number of solutions(…)
But upon more mature reflection the position held by hypothesis was seen; it was recognised that it is as necessary to the experimenter as it is to the mathematician. And then the doubt arose if all these constructions are built on solid foundations. The conclusion was drawn that a breath would bring them to the ground. This sceptical attitude does not escape the charge of superficiality. To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection
Instead of a summary condemnation we should examine with the utmost care the rôle of hypothesis; we shall then recognise not only that it is necessary, but that in most cases it is legitimate. We shall also see that there are several kinds of hypotheses; that some are verifiable, and when once confirmed by experiment become truths of great fertility; that others may be useful to us in fixing our ideas; and finally, that others are hypotheses only in appearance, and reduce to definitions or to conventions in disguise. The latter are to be met with especially in mathematics and in the sciences to which it is applied.(…)
Our laws are therefore like those of an absolute monarch, who is wise and consults his council of state. Some people have been struck by this characteristic of free convention which may be recognised in certain fundamental principles of the sciences. Some have set no limits to their generalisations, and at the same time they have forgotten that there is a difference between liberty and the purely arbitrary. So that they are compelled to end in what is called nominalism ; they have asked if the savant is not the dupe of his own definitions, and if the world he thinks he has discovered is not simply the creation of his own caprice. Under these conditions science would retain its certainty, but would not attain its object, and would become powerless. Now, we daily see what science is doing for us. This could not be unless it taught us something about reality; the aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relations between things; outside those relations there is no reality knowable.(…)
Space is another framework which we impose on the world. Whence are the first principles of geometry de- rived? Are they imposed on us by logic? Lobatschewsky, by inventing non-Euclidean geometries, has shown that this is not the case. Is space revealed to us by our senses? No; for the space revealed to us by our senses is absolutely different from the space of geometry. Is geometry derived from experience? Careful discussion will give the answer—no! We therefore conclude that the principles of geometry are only conventions; but these conventions are not arbitrary, and if transported into another world (which I shall call the non-Euclidean world, and which I shall endeavour to describe), we shall find ourselves compelled to adopt more of them.
From Yuri Manin’s – Mathematics as Metaphor:
Note: that this PDF is an OCR copy of the original, this might have allowed point mutations to have crept in that further enrich the text?
What is relevant, is the imbalance between various basic values which is produced by the emphasis on proof. Proof itself is a derivate of the notion of “truth”. There are a lot of values besides truth, among them “activities”, “beauty” and “understanding”, which are essential in the high school teaching and later. Neglecting precisely these values, a teacher (or a university professor) tragically fails. Unfortunately, this also is not universally recognized. A sociological analysis of the controversies around the Catastroph Theory of René Thorn shows, that exactly the shift of orientation from the formal truth to understanding provoked such a sharp criticism. But of course, the Catastroph Theory is one of the developed mathematical metaphors and should only be judged as such.
Pedagogically, a proof is just one of the genres of a mathematical text. There are many different genres: a calculation, a commented sketch, a computer program, a description of an algorithmic language, or such a neglected kind as a discussion of the connections between a formal definition and intuitive notions. Every genre has its own laws, in particular, laws of rigour, which only are not codified because they were not payed a special attention.
A central problem of a teacher is to demonstrate at the restricted area of his or her course the variety of types of mathematical activities and underlying value orientations. Of course, this variety is hierarchically organized. The goals may vary from achieving an elementary arithmetical and logical literacy to programming skills, and from the simplest everyday problems to the principles of modern scientific thinking. In the spectrum of these goals, the emphasis on the norms of “rigorous proof” can safely occupy a peripheral position.
But having said all this, I must stress that my argumentation by no means undermines the ideal of a rigorous mathematical reasoning. This ideal is a fundamental constituting principle of mathematics, and in this sense Bourbaki is certainly right. Having no external object of study, being based on a consensus of a restricted circle of devotees, the mathematics could not develop without the permanent control of rigid rules of game. Applicability of mathematics in the strict sense of this word (like its indispensability in the Apollo project) is due to our ability to control series of symbolic manipulations of fantastic length.
The existence of this ideal is far more essential than its unattainability. The freedom of mathematics (G. Cantor) can only develop in the limits of iron necessity. The hardware of modern computers is an incarnation of this necessity.
Metaphor helps a human being to breathe in this rarefied atmosphere of Gods.
[EDIT: I have been asking myself a few (hopefully?) guiding questions lately: e.g. To what end? Now that there are results, what to do with them? What would XXX do? (Where XXX is not a blue movie but rather your favorite totally morally sound deity or corporeal yet morally awesome human.) To the first question and the last, there might be a beginning of an answer. Consider the following thought experiment: You are you. Your life has been life-cribbed and your personhood hacked/cracked. Forget the technological stuff for a second, leave that to the beards. Instead ask, “Is there anything interesting in my experience that might provide some intuition or knowledge that I (you) can use to do something useful and interesting?” For example, you (being you) think about your experience, that is, your life (cribbed) and your personhood (cracked?) and ask if there is anything new here going on? People have been doing hacks and cribbing in various ways for a long time. Is there anything new going on or is the experience you’ve had simply a manifestation of age-old crap e.g. gossip translated to new media (think radio on the internet not TVOTR (the latter being quality not crap and, while not exactly new, not really age-old either)? You read the following remarks (Manin) from the AMS interview linked below: “For me, this story marks the period in which mathematics and physics parted ways. This divergence continued until about the 1950s. The physicists dreamed up quantum mechanics, in which they found a need for Hilbert space, Schrödinger’s equations, the quantum of action, the uncertainty principle, the delta function. This was a completely new type of physics and a completely new type of philosophy.” and realize that interesting new work is sometimes (always?) attached to people noticing and carefully thinking about interesting or novel situations (or ideas (that’s the part that might make the always? go) (e.g. physicists dreamed up quantum mechanics, in which they found a need for Hilbert space). Now without trying to draw a parallel to your experience and the “dreaming up of quantum mechanics” but recognizing it as (at least somewhat) unique, you do what you pretty much always do (if you were me) and poke around on teh intarweb for something to help you (me) make sense out of our experience. Upon poking, you have some ideas…
Another question you could ask yourself if you found your life fragged is, “Is there any interesting aspect of the structure underlying this sort of thing that might be worth reading some intelligent person’s assessment of?” Well, luckily (for you?) there might be a paper online at the arXiv that talks in straight language so that even I can understand it (you, being you, could easily read more advanced material on such things). Thinking over your experience of participating in deference (or, I shake my head and sigh as I have participated in something like deference (2, 3) but that lacks a key element from the previous definition. You, of course, being you, have never taken the respect out of deference and participated in such questionable behavior. I being me, unfortunately, know both sides of the whole issue of the above and have begin to take steps to modify my behavior based on new data and experiences (this, experts tell me is something called learning, an activity I am still struggling to learn.) and… (popping the stack) fashionable nonsense aka fashionista-ism.
You (being me) are sad that you have wasted time in such ways (submission, fashionable nonsense) and would like to make up for it by adding value. (This is because an influential person in your life once told you that this is something that you should do and you defered.)
Having performed our poor men’s thought experiment on the cheap. Now I/you/we are going to go a) watch TV OR b) read Aldous‘ “Interacting particle systems as stochastic social dynamics“? (I like (b) (and also the second link in add value above) but I wouldn’t bet on it being the likeliest choice unfortunately. Things being pretty unsurprising around here.)]
Note: the below license obtains for this page but not the images (go Google your own images of famous mathematicians), nor the linked-to material, nor the author’s life and the life of the author’s family and their living space, nor sounds emanating from said living space, nor other aspects of the author’s life or that of his family (unless specific permission is given by the author or at least a reasonable conversation with the author, or even a the slightest effort at communicating with, getting to know or otherwise having a relationship (traditionally construed) with the author, or perhaps a discreet email or cup of coffee with the author, … , etc.). Note, despite the general “La mort de l’auteur”-ness of the present day, people are still people (yes, even ones that have become post-human by being fragged into a million-million tiny pieces and sent on their way to information heaven), with eyeballs and ears (usually) attached to brains that process information and that can perform tasks like sitting at their computer, writing a pastiche-piece about some interesting philosophy of math/science that they just read, hearing the TV from the other room, rolling their eyes (vision science), getting up to investigate (locomotion science)…
Thanks again to everyone for all their patience with my incredibly slow learning rate and other assorted difficulties and defects.
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