“math”
23 posts under this tag.
Game: 2 players take turns to say a number between 1 and 9. Numbers may not be repeated. The goal is to be the first to say 3 numbers which add up to 15.
Sounds like fun? Try it with a friend!
Fun it ain’t.
It’s hard to remember the said numbers and “playing” is a chore involving many additions in your head. Maybe it’s fun for the better short-term memory endowed or those who enjoy arithmetic but that ain’t me.
Turns out that game above is none other than the beloved tic-tac-toe. You see:
This is what I love about information design (and what I tried to do in my calendars) this is its art, its magic: it can turn a chore into a game! It recasts our weaknesses —linear, verbal processing— into a form suitable for our talents —gestalt visual processing.
In math words: it finds useful language-graph same-shapes (isomorphisms)!
Where, but the web, would you find someone like Oliver Steele? This ain’t no metaphor. That name was a link. I’m not talking about Oliver Steele the person, I haven’t met him (though I apparently am 1-degree of separation from him; weird, that). I’m not talking about the sweating, walking, pinchable, space-and-time-and-flesh-bound avatar, I’m talking about his online persona. And either I’ve gotten crazy enough or technology has advanced enough that I’m ready to treat Oliver Steele —the link, his blog, words, diagrams, code, and further media— as a person by its own merits.
And, boy, is he an interesting guy:
I studied math in college because I didn’t believe it. Never could understand how or why someone would come up with the stuff we were being teached. Thanks to some innate verbal ability and motherly discipline, I was thankfully “good” at it though, good enough to realize that what we were “learning” was nothing but mindless regurgitation.
“The Humean predicament is the human predicament”
What are you absolutely certain of? Of what are you sure without any conceivable doubt? What is true no matter what? What is necessarily true? Just one thing. Whatever. As long as you’re sure.
I’ve been playing the game for a while and I’ve been shocked to be unable to answer the question. Now, of course I’m familiar with Hume’s skepticism (you don’t really know an apple is going to fall, you’ve just seen all similar objects fall before at similar conditions but you don’t know) and I thought I knew how dear truth was but lately, slowly, I’ve started to realize that not even reason or logic are to be trusted.
Let’s start by quickly demolishing every statement about experience, like, say, that you are, well, you, that you broke your knee when you were fifteen, that your mother exists, that other people exist (solipsism). The usual shortcut is just to ask you how do you know it isn’t all a dream, but I prefer Russell’s more imaginative version, the extreme omphalos hypothesis: how do you know that the world wasn’t created five seconds ago, set in motion, and with fake memories? Clever, huh?
OK, that sweeps off a good big swath of possible answers. As for reason/logic, its problem is that it’s either redundant or not binding at all. But don’t 2 + 2 = 4 whatever fucking nightmare the world might turn out to be? How could time or space not exist? My gosh, can you look me in the eye, and tell me that numbers aren’t infinite? How demented do you need to be to doubt Aristotle’s syllogisms, the very rules of thought (if it’s true that humans are mortal and that Socrates is human, Socrates has to be mortal!)?
But it turns out these conceptual statements aren’t certainties either. When you probe them further, carefully, rigorously, you realize that to advance you have to start defining. If you do it conscientiously, defining or making explicit even the dumbest, most-taken-for-granted assumptions you start to realize that 2 + 2 = 4 because you said so, because you assumed your conclusion from the get-go, and your statements are true in the same empty way that a bachelor can’t be married or a car has to be an automobile too. Yes, it’s a kind of truth, but a rather measly one.
The other thing that usually happens when you probe concepts is one of the most wondrous experiences I know of, exhilarating and unnerving at the same time, dizzying. I call it sense of could. It means taking an entrenched concept and realizing it is not necessarily so, discovering your singularity is just an instance of something subtler, deeper, finding out your rose is one among thousands, seeing that what you thought fixed is just another degree of motion.
Like when Cantor found out there are many kinds of infinities, some bigger than others (!). Like when you realize logic isn’t the complete science Kant thought and open the gates to the non-classical logics. Like when you probe the very fabric of the universe by looking for primitives to space and time. More worldly, like when you question your ethics, your religion, your politics, and you find only possibility where you were looking for necessity.
Now, those two options, redundancy and non-necessity, are the ones I’ve always stumbled upon but I don’t really know that happens for every concept. Or neither do I know if you can dismiss all experience in one fell stroke. That is, I’m, of course, not even sure that you can’t be sure of anything. Would you care volunteering an answer? %(p)Or a question?)%
Charles S. Peirce has been called by Britannica “the most original and the most versatile intellect that the Americas have so far produced.” Bertrand Russell considered him “one of the most original minds of the later nineteenth century, and the greatest American thinker ever,” and Karl Popper goes all out, seeing him as “one of the greatest philosophers of all times.”
I just met him a couple of weeks ago and I couldn’t be more impressed: the man’s a fricking genius, practically inventing semiotics and modern logic, making major contributions to the philosophy of science and epistemology. I would remember him forever just for his offhand naming of math as the “hypothetical or conditional science.” (the could science? the moot science?) and I have the sneaking suspicion that ours will be a lifelong acquaintance.
How not to be intrigued by a man who could explain reason in a sentence?
For reasoning consists in the observation that where certain relations subsist certain others are found, and it accordingly requires the exhibition of the relations reasoned within an icon.
OK, to fully get the above quote you should be familiar with Peirce’s brilliant and influential classification of signs into ”icons, which signify by virtue of resemblance [think painting], indices, which signify by virtue of a physical connection with the object [think weathervane or tally], and symbols, which signify by virtue of the existence of a rule governing their interpretation [think words].”SOURCE
Then there’s Peirce “discovery” of abductive reasoning, the third major class of logical reasoning and for which I’ve found no better (or shorter) intro than the logical reasoning pedia.
And to finish this Peirce appetizer you must check out Peter Skagestad’s Thinking With Machines article. He gives a summary of Peirce’s semiotic to make a most intriguing comparison with the thought of human intelligence augmentationists like Doug Engelbart ELZR. Fascinating stuff really.
A math experiment was carried out recently when Alex Smith —an Electronic and Computer Engineering undergraduate with “a background in mathematics and esoteric programming languages”— proved that the Turing machine below is in fact universal, making it the simplest universal Turing machine possible. In other words, the cute graph below are the instructions for an abstract symbol-manipulating machine that can in principle do anything your computer (or any other computer for that matter) can do.
Stephen Wolfram, who made the conjecture and offered a $25k reward for proving it, reports:
We’ve come a long way since Alan Turing’s original 1936 universal Turing machine—taking four pages of dense notation to describe.
We did an experiment; and PCE [the Principle of Computational Equivalence] was validated.
But unlike some science experiments, it didn’t take a multibillion-dollar particle accelerator. It just took a 20-year-old undergraduate with a PC.
[It’s] a wonderful monument in the computational universe—a marker at the edge of universality for Turing machines.
It’s a very satisfying way to spend $25,000.
Now, ain’t this just breathtaking?
“I’ve always accepted my groggy mornings because they come with my glorious nights.”
anglo latin; the origin of latin america as a word; my high school wasn’t bilingual, it was English with Spanish as a second language; I always say math and language are the same ability because they’re the same thing, math is just specialized talk about numbers; but then zoology is just talk about animals…; the difference is that math is, in a farly unparalleled way, advanced through language itself (as opposed to, say, zoology, which, advances chiefly through observation); the way forward in language is through logic; logic is syntax; understanding is synthesizing and paraphrasing; {an algorithm just like flooding to create a machine that understood, as previously defined—one wouldn’t even need to define what a word is}; Wikipedia syntax highlighting!
Glorious, thought-drunken night.
Why do we call something a “number”?: Well, perhaps because it has a “direct” relationship with several things that have hitherto been called number; and this can be said to give it an indirect relationship to other things we call the same name.
And we extend our concept of number as in spinning a thread we twist fiber on fiber. And the strength of the thread does not reside in the fact that some one fiber runs through its whole length, but in the overlapping of the fibers.
Ludwig Wittgenstein, Philosophical InvestigationsEEM
Always have loved them. Always have obsessed about them. I treasure my favorites and revisit them again and again—I could barely think without them. I have a tag for them in this blog (here) and I almost started “a collection of beautiful definitions” to go with my eemadges website (“a collection of beautiful descriptions”). A good definition more than justifies a whole book. A good book always has many good definitions in it. Good people always carry several good definitions with them—you just have to know how to tease them out.
And yet I seem to get into all kinds of tiresome, silly discussions when I try to share them with friends. Besides my not to be belittled incompetence as an explainer and my fabled monomanias, I believe a basic misunderstanding regarding their nature is at the heart of the matter.
You see, most people seem to never have moved over the idea of a definition as distilled truth—the one true essence which both captures everything that should be captured and leaves nothing that shouldn’t be left out. Definitions as platonic ideals—the perfect divine forms of which we only see shadows. The one golden fiber that runs trough all the thread.
The problem with this view, of course, is that it is crippling in its obsession with perfection. It intimidates and nurtures ridiculous expectations. If we had had to delay mathematics until we had a “perfect” definition of number we would still be waiting.
In their supposed perfection, definitions only become cages. And we easily get to the point when not only it isn’t believed that things like “love”, “mind”, “conscience”, or “happiness” could ever be defined (again, as if there was one true definition to rule them all), but the very possibility is viewed with dread. Dread that what once was magic and alive is cramped and crippled into a cage.
A much more interesting view of definitions, in my opinion, is to regard them as tools for thought, and as such, to value them on their usefulness and pick the one appropriate for the task at hand—platonic truth is only one of the many, many things we can ask of them. Most importantly, we ought to recognize that we need them—a brain unaided can do only so much. Thinking without them is like hammering with your bare fists—it’s painful and ineffectual. Yes, they are only one (verbal) kind of tool and we run the risk of starting to see everything as a nail, but they are still one of the most basic and powerful tools we have and they have so far been needlessly feared and vilified.
Definitions are semantic flashlights, casting light on some meaning corners, shadow on some others. That everything be alight is only one criteria (ultimately impossible; only emptiness can be shadelessly illuminated), there are others—that it be bright, that it be dim, that it illuminate (or obscure!) a particular patch, that it be pristinely white, that it tint its subjects with its color, that it be diffuse, that it be focused, that it be favorable, that it be unfavorable… We say, teasingly, that an American is a “man with two hands and four wheels” not because we believe that it happens to be a perfect embodiment of what it means to be an American, but because we believe it casts them in an interesting light.
So the effort to define “play” or “capital” or “freedom” is not to pin the butterfly down and put it in formaldehyde, it’s to find new ways to look at it, new sources of joy and understanding. Definitions do not diminish their subjects, they reveal them.
Of course! A 3-legged table wouldn’t wobble! Why? Any three points define a plane. Why did I never think of that?
So you’ve purchased your coffee and chosen to sit at one of those round outdoor tables. As you lean on the table to write comments on a paper, it rocks annoyingly, possibly spilling some of your coffee. You try moving the table slightly on the uneven pavement, hoping to stumble into a stable configuration for its four feet, but several attempts fail. Eventually you resort to shimming one of the table feet with a piece of folded up paper, or a stack of sweetener packets, and this creates at least a metastable condition. Looking around, you notice that many other tables have similar combat repairs, so that the cafe looks like a furniture trauma ward.
Why don’t these tables have three legs instead of four? With three legs, they wouldn’t rock on uneven surfaces because any three points define a plane. You wouldn’t need those adjustable table feet that no one ever bothers to adjust because it’s so awkward to lean down and twist them. While each leg would have to be slightly bigger, you’d have fewer assembly or machining steps to perform. Is a 60 degree angle that hard to produce in this day and age?
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