Mathematica by David Bessis

[GeneralChaos]

@Subrosian Smithy

Thanks for posting those quotes from Mathematica in @Savestate's thread! I started reading it tonight.

I just read the bit about the Patience game and describing shapes to people who don't have any concept of them. It's bad! It's failing to account for the fact that block puzzles aren't made of abstract 2d shapes. These hypothetical people are going to use the method he described on a triangle block and go "Point, point, point, point. Hmm, must be a 'square.'" And then, if it doesn't just go in the square hole, it will simply not work! Either way, they will not grasp the important intuition about shapes! It is a good example of how hard to teach mathematical concepts are, but it's just so ugh.

He could have attempted to describe a technique for orienting the blocks. For instance, press your palms together. That tactile sensation where they meet is flat, and your palms are parallel. (insert 2k words on the definition of 'parallel") In order to get meaningful shape measurements, you start by finding the two flat sides which are parallel by holding it between your palms. Then transfer it to one hand, with the thumb on one of the parallel faces and the fingers on the other. Use your freed hand to measure the signature of the block.

I definitely get why he wouldn't have included the extra stuff even if an editor pointed it out, but goddamn does it bug me.

---

It's funny that annoyance gets me to make a thread where sincere enjoyment of the book didn't. I really like Bessis' writing.

Early on, when he prompted me to think about getting a straight line to intersect with a circle in three places, I played with it for a while. I thought about a spherical geometry, and tried to get a straight line to intersect with a circle by spiraling around, before I realized that that's not right. My mental image of the spiraling line first expanded to cover all of the 2d spherical geometry, then abruptly collapsed into a circle. A straight line in a spherical geometry is just a great circle! I did get the last laugh, though! I can get a straight line to intersect a circle infinitely many times if it's in a spherical geometry! It's the same circle!

I kept coming back to that experience as I read the book. It's a great toy example of the kind of mental images he's talking about. Much of the book hit harder with that experience so close to hand. Grothendieck's bit about just getting hands on with the mental objects and seeing what happens in particular.

I certainly wouldn't call myself a great mathematician. I did study enough math that I've gotta know a least a bit, though. As I read the book, I kept going "Yeah, that's like the time I did that one thing!" I feel like I blindly stumbled into a lot of useful intuitions, and also missed a bunch of opportunities to learn better.

I remember this time just before eighth grade astronomy, where I was looking a poster with some differential equation for orbital mechanics. I applied the rules of pre-calc simplification. Of course, in pre-calc all the variables are x and y, rather than things with subscripts, so I just treated the subscripts as plain numbers! I also canceled out all the d's and δ's because clearly that's the same variable, right? I showed this to a classmate, who was sure what I did was wrong, but could not explain it to me before class. That was a missed opportunity. I could have learned about subscripts and operators, about the notation of physics. On the other hand, the thing I came up with bore a striking resemblance to a unit analysis. I had canceled the d's, which left time in the denominator. I don't still have that piece of paper, but it would be fun to see how I went wrong and how close I was to something worth thinking about further!

Anyway, I want to get back to reading the book.

 

[GeneralChaos]

I'm in the chapter where Bessis rags on Kuhnman for promoting System 2 in opposition to System 1.

I did the "Add the numbers 1+2+3+...+97+98+99+100" differently than the example. I basically decided that 1+99 and similar sums are equal to 100, and 100 is a nice number to work with. Then I noticed that 1 and 99 are both odd, so when I folded the numbers [1,99] in half to make a bunch of pairs which sum to 100, I'd have a left-over at fifty, because once 49 and 51 sum to 100 there's nothing to add to 50. I had basically grabbed the number line at 50, folded it in half, and zipped it together. I was left with 49 pairs that summed to 100, an unpaired 50, and the 100 on the end. That is to say, I had 50 100's and an extra 50: 5050.

This took seconds rather than minutes, and I did it in my head rather than on paper, so overall it was a success. I'm glad that I came up with a different trick than the one Gauss did. Feels good, man. I really do get what he meant with:

In mathematics, the sudden occurrence of a miracle or an idea that seems to come out of nowhere is always the signal that you're missing an image. Your way of looking at things isn't the right one. Something is missing. There exists a better way, simpler, clearer, deeper, that you don't know yet and that, perhaps, no one yet knows. Looking for and finding the right way of seeing things is the driving force of mathematics. It's the main source of pleasure you can take from it.

Each time someone talks to you about "tricks," they're telling you to stop thinking at precisely the moment when it starts to get interesting.

Click to shrink...

 

[Subrosian Smithy]

I certainly wouldn't call myself a great mathematician. I did study enough math that I've gotta know a least a bit, though. As I read the book, I kept going "Yeah, that's like the time I did that one thing!" I feel like I blindly stumbled into a lot of useful intuitions, and also missed a bunch of opportunities to learn better.

Click to shrink...

I've been having the same experience, lol. A few months ago in chemistry class we were asked to identify the number of transitions between electron energy levels in an atom with some number n of discrete energy levels, which were depicted in a figure as rungs on a ladder. It wasn't a particularly large n, and I had a ruler, so I instead of scribbling on the ladder figure, I immediately drew a rough n-gon, drew all the rest of the lines that could go between all of the vertices, and found the correct answer by counting those up.

I assumed at the time that I'd read about this analogy to simplex graphs somewhere, like in a book of recreational mathematics. But I've recently been going through my old papers and found scratch notes I must have taken while I was bored in middle school geometry class. I'd already performed the procedure myself for all values of n between one and eight or so, and even worked out the general formula for the number of edges in a simplex with n vertices. (Cute.)

I also learned recently while working on a bunch of stoichiometry problems and asking a senior for help that it was possible to do stoichiometry with basic linear algebra, even though I didn't understand why. But I had another visualization eureka moment from my old memories of doing arithmetic on the complex plane and realizing you could do arithmetic on any ordered pairs - the reactants and products in a chemical reaction are just vectors! Each atomic element is a basis axis, and by assigning coefficients to the different molecules and compounds on each side of the chemical reaction, you're arranging sets of vectors to reach the same point at the smallest non-trivial distance from zero.

I'm actually a little sad we haven't covered the linear algebra methods in class. I understand why we wouldn't, lin-A isn't a primary math prerequisite for this class and we have a lot of ground to cover, but it seems elegant and general. I think the professor might even have been asking the other day what method I was using for stoichiometry - I kind of mumbled her off, because I assumed she was asking about an unfinished problem I left in the margin of my notebook when she came around to check our lab reports. (I went overboard trying to algebraically demonstrate how one quantity would vary in proportion to another quantity in an experiment when I could have reached the same answer 10x faster through intuition alone, and I got embarassed.)

I keep wondering if I should send her an email to explain myself, but I'm probably overthinking it.

 

[GeneralChaos]

I've been having the same experience, lol. A few months ago in chemistry class we were asked to identify the number of transitions between electron energy levels in an atom with some number n of discrete energy levels, which were depicted in a figure as rungs on a ladder. It wasn't a particularly large n, and I had a ruler, so I instead of scribbling on the ladder figure, I immediately drew a rough n-gon, drew all the rest of the lines that could go between all of the vertices, and found the correct answer by counting those up.

I assumed at the time that I'd read about this analogy to simplex graphs somewhere, like in a book of recreational mathematics. But I've recently been going through my old papers and found scratch notes I must have taken while I was bored in middle school geometry class. I'd already performed the procedure myself for all values of n between one and eight or so, and even worked out the general formula for the number of edges in a simplex with n vertices. (Cute.)

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That's an interesting approach!

After working through your method (and noticing that with every extra point you add, you get one more edge for each preexisting point!) and figuring out the formula for

\[E_{n+1} = E_n + n = \sum_{i=0}^n i = \frac{n(n+1)}{2} \]

I've gotta admit, my method feels unpolished. It got the job done, but had loose ends which I handled separately. I certainly didn't generalize from the specific example to all the relevant sums until you prompted me to.

On the upside, I did it in my head and got it right the first time. The loose ends were obvious and easy to clean up. If you've gotta have loose ends, at least make sure they're noticeable and take care of them!

---

In chapter 18, subsection Taking Words At Face Value, Bessis challenges us to define our tolerance margin for the perceptual definition of a sphere. I don't have a tolerance margin; it shifts depending on the practical situation. Honestly, when it comes to accurately talking about human language definitions I often use the concepts introduced to me by The Categories Were Made For Man, Not Man For The Categories and the references it links.

What are some pertinent features of spheres? Well, they're round. They roll around, in any direction (unlike cylinders, which can only roll perpendicular to their axis). They are a shape of undisturbed surface tension, too. A floating soap bubble is a sphere, even if you blow on it and it wiggles a little. Even though you couldn't roll it without it breaking, it's still got the shape that is conducive to rolling, much like the Earth.

Spheres are convex. If you hold a toy poké ball in your hand, even if it's got a flat bit at the bottom so it doesn't roll all the time, it's still got so much convex character! Even if you open it, so you've basically got two hemispheres on a hinge, I'd still call it a poké ball without caveats like poké-two-hemispheres.

I've got a Newton's cradle; I'd call the weights in it spheres, even though they've got little attachment tubes on the top that keep them from spinning on the strings. If the deviation from mathematical spherehood is primarily a bit of connective tissue, it's still getting called a sphere.

This isn't particularly coherent, but it definitely feels like I've thought about how fuzzy the borders of human language categories are, even though I gave up on exercise he actually described immediately. I probably could turn it into a nice criteria-based definition, but I want to move on.

 

[GeneralChaos]

Article:

Notes and Further Reading​

Chapter 3. The Power of Thought​

In this chapter, the ability to "see" a circle in one's head is presented as a universal human capability, which isn't entirely correct. In 2015, a research team led by Adam Zeman at the University of Exeter described a rare condition, aphantasia, characterized by the inability to create mental imagery. A 2022 study estimates the prevalence of complete aphantasia at around 1 percent. See https://en.wikipedia.org/wiki/Aphantasia for details and references.

When writing this book, I was faced with the challenge of describing what's going on inside our heads. It was natural for me to put an emphasis on the visual experience, as it is easy to convey and most people find it striking. My apologies to readers with aphantasia, who may occasionally feel left out.

One reader with aphantasia reached out after the original edition was published, and I had the opportunity to discuss this chapter with him. While he can't "see" the line sweeping across the circle, he does find "obvious" that a line cannot intersect a circle in more than two points (without being able to provide a reason).

This lone testimonial is by no way a robust scientific study, but it does help illustrate a key point that is reasserted in chapter 16: mathematical intuition comes in many shapes and forms. It doesn't have to be visual.

I'm aphantasiac, and I'm really glad this endnote was included.

When I imagine a parabola, I don't see something in my mind's eye. Instead, I remember the motor sensation of my eyes moving to trace along a parabola. Often my eyes will actually move along the parabola! I can do similarly for just about anything I imagine; when I was imagining a spiraling line around a spherical geometry, my eyes were moving. To the extent that I had a mental image, it was more like a colorless wireframe. I couldn't see it any more than I can see the air in front of my face.

I can move the point of my attention without actually moving my eyes. I can even split it into multiple points, if for instance I wanted to imagine a paraboloid. If I were to gesture the motion that's happening in my head, I'd hold my hand in front of me with my fingers together. Then, as I raise my hand, I'd spread my fingers apart such that they traced along the paraboloid. It's not the same, because the points of attention in my head aren't only five in number, but it's close enough.

I also often do partial derivative or volume integral things. I'll trace my attention around a circle and then scan that circle up and down to imagine a cylinder, for example. I don't see a cylinder. I just imagine my attention sliding along it.

 

[Subrosian Smithy]

I also often do partial derivative or volume integral things. I'll trace my attention around a circle and then scan that up and down to imagine a cylinder, for example. I don't see a cylinder. I just imagine my attention sliding along it.

Click to shrink...

Woah, that's amazing. But it makes perfect sense! Of course geometric representations are multiply realizable!! Duh!!!

I always used to think it was a stupid question to ask (or a question that needed to be dissolved), but Bessis' playful realism about mental representations really makes me think there's a meaningful way to think about things like "is your red the same as my red?"... I wonder just how many neurodivergences hinge on specific layouts of representational structures within the brain, and whether I should take anything from that for how I work within my disabilities in the future.

I didn't post it in the other thread, but the parts of the book where Bessis talks about the ~dark side~ of mathematics 100% reminds me of how all the hardcore internet Buddhists discuss the risks of insight meditation, to boot. (Yes, you can unlock your hidden potential, rebuild your consciousness, see into the fifth dimension, and experience transcendence, but you might also wander into the wrong attractor basin, give yourself qi deviation, and explode!)

 

[GeneralChaos]

Woah, that's amazing. But it makes perfect sense! Of course geometric representations are multiply realizable!! Duh!!!

I always used to think it was a stupid question to ask (or a question that needed to be dissolved), but Bessis' playful realism about mental representations really makes me think there's a meaningful way to think about things like "is your red the same as my red?"... I wonder just how many neurodivergences hinge on specific layouts of representational structures within the brain, and whether I should take anything from that for how I work within my disabilities in the future.

Click to shrink...

Yeah, it blew my mind when I consciously realized that the eye motion thing was what I was actually doing when I was preparing to draw a parabola!

After reading the book, I've decided to try to improve my imaginary object permanence. It isn't really a thing I've tried before. I've currently got an imaginary cylinder just floating about a foot above my desk, off the left side of my monitor. I'm trying to be consciously aware of it the way I'd be aware of, say, the car's hatchback a few inches above my head while I bend over to pick up something inside. I wonder if it'll still feel like it's there in the morning!

 

[Subrosian Smithy]

After reading the book, I've decided to try to improve my imaginary object permanence.

Click to shrink...

Godspeed!

Right now I'm looking to see if I can't whip up some RNG widget to produce practice problems for me automatically. I want to brush up on my arithmetic.

 

[Subrosian Smithy]

I never really was interested in numbers, and never spent enough time thinking about them. That being said, through repetition and habit, I did manage to develop a degree of familiarity with them. I understand them up to a certain level. But I've never felt myself capable of being creative in number theory or in arithmetic. Like a tennis player whose backhand is relatively weak and who shifts to play his forehand, I've fallen into the habit of avoiding numbers whenever possible, and looking for geometric interpretations of quantitative statements.

If my primary interest had been with numbers, I would certainly have developed a much stronger personal bond with them. This intuition might have been primarily visual, or of an entirely different nature. In the end it doesn't matter that much. All mathematicians approach mathematical objects intuitively, but intuitions come in many shapes and forms.

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I've been paying extra attention to my thought process while doing arithmetic in chemistry class to see exactly what kind of mental representations pass through my mind. Trip report.

While doing arithmetic on paper, I cannot "see" large numbers - those numbers with more than three non-zero digits, or with more total digits than can be subitized - unless they are delimited into groups of three digits, as is the difference between '123456789' and '123,456,789'.
I still "feel" a sense of magnitude for non-delimited large numbers, but only in the length of each string of digits, rather than as a base numerical property, and I do not apprehend them as integers or rational quantities. (They are pure language games.)

While manipulating delimited large numbers on paper, I can apprehend the digit strings as proper quantities, but their associated mental image is slightly 'laggy'. It is as if I'm following a road map at the speed of thought in order to reach an address (on the number line?) that I have heard of in the past but have never previously visited in person.

The very smallest natural numbers form an oasis of stability among other images and representations. They are exceptionally immediate and concrete; generally, the smaller the natural number, the more palpable its own representation and evocation within my mind.

Zero is stereotypically elusive. It has no synaesthetic character save for an absence of character while read as a placeholder digit, addend, or subtrahend.
Applied as a multiplier, zero feels vaguely like a trapdoor, a one-way gateway, an energy level collapse, or a fall into Unbeing. Zeroth powers feel similar, but with somewhat less terminal finality (because they output one rather than zero).
Applied as a divisor, zero feels vaguely like a one-way gateway I'm already on the wrong side of - like actual infinity is supposed to be just across the turnstile, but I can't ever reach it in reality because I'm already stuck inside this stupid baka event horizon.
In the coordinate or complex plane, zero feels very strongly like a wheel. As if the rounded '0' symbol is a synecdoche for the rotations which can occur around the axis of rotation at zero.
On the number line, zero feels like a fulcrum, a center of gravity, or a bidirectional doorway.

One is extremely tangible, but also too fundamental and immanent to subdivide into lower-order units of meaning. It fluctuates in character under my attention between a dimensionless point, a monad which can encompass or enclose all other numeric structures, and an ontological root from which the meaning of all other numbers is outwardly emanated - zero's far more palpable and ontologically compelling big-sis onee-san.

Small numbers greater than one are strongly geometric. Two feels like two dots right next to each other in my minds eye. It may also include an implied edge drawn between them to represent their association within a single structure, like a q-tip or a diatomic atom. If I'm not careful, the version of this image with the edge drawn in "resonates" very strongly with concepts of polarity (if the two regions of interest are at either end of the line) or duality (if the two regions of interest are across 'either side' of the line).

Three is weakly triangular, or at least, weak relative to other small integers. It is mostly non-visual and subitive, but it's easy to summon triangular representations of three dots, and any triangular 'three' I draw in my mind resonates vaguely with the boundary between one-dimensional and two-dimensional sets of points. (I suspect I would find three more profound if I were raised Christian.)

Four is very strongly square, rather than generalizing the triangle to a tetrahedron. It is four dots in a regular square, which geometrically resonates with the four cardinal directions and the four spokes of the cartesian plane (and the division of an area into four quadrants by two axes).

Five is weakly pentagonal. It feels "more prime" than three does, for coming after a demonstration of four's divisibility, but not that prime. Because five and two are the only non-trivial factors of ten - and because I have five fingers on each hand - five is much more sharply apprehensible as a segment in a set of ten or a halving of the radix than a number in itself. Because it is the first number with non-trivial remainder after division, five also feels weakly as if it has a hook or cusp trailing out of it, like a two-by-two girl with something extra.

Six is the first number with non-polygonal geometric representation. I can apprehend it strongly as a hexagonal group of points, but without a problem case to segment a circle into six congruent angles, the vertices greatly want to "collapse" into a group of three points by two points. Interestingly, although six only has two non-trivial factors and multiplication is commutitive, six already demonstrates 'resonance structures': it palpates differently depending on whether the two-by-three points are laid out horizontally or vertically in my field of vision, as I 'read' the structure from right to left.

Seven is weakly heptagonal. The prime numbers greater than two are all only weakly geometric in my mind; as with five, seven and onwards feel weakly hooked or cusped, but not particularly.
Interestingly, although they're only weakly geometric and would be impossible to tesselate even if they were, the small prime numbers from five to eleven all feel abstractly elegant in my mind's eye, oscillating between a pleasantly rounded bouba and a symmetrical spoke kiki. Thirteen and three are edge cases on the boundary conditions of this oasis; other prime numbers are ungainly, and jitter between lumpen potato bouba and jagged glass kiki. I suspect this aesthetic phantasmagoria is related to the range in which I can effectively subitize prime numbers of objects.

Eight is weakly octagonal. It is the last natural number with a natural and polygonal geometric representation in my mind, and like six, it very strongly wishes to collapse into grids of points, to the point that I will not summon octagons without being prompted by context. Like six, eight also has resonance structures, but more complicated ones: it vibrates between two-by-four, four-by-two, and two-by-two-by-two, either rotating, folding up, or unfolding at the speed of thought within my mind's eye. As with four's propensity to organize as a square, when not unfolded into smaller pieces, eight feels quite prototypically cubic.

Nine is a strong and perfect square of three points by three points. Because of its close proximity to the decimal radix, nine also feels weakly slotted, with an empty space along its perimeter where a hanging hook or cusp could take her hand to complete a set of ten. All numbers between one and nine feel similarly paired, such that one goes with nine, two goes with eight, three goes with seven, four goes with six, and five goes with itself. However, these latter relationships are not as palpable as nine and one.

Numbers of ten and above are far less geometric, with further falling-off such that they become increasingly sparse quantity-signifiers between twenty and forty. I cannot effectively summon or manipulate dot grid representations for the factorization of numbers over forty or so - the quantities involved are too large, and the possible resonance structures too indistinct and incessant. However, I do not think this is a fundamental limitation in the slightest - I can still summon the first twelve perfect squares and the first five perfect cubes as geometric signposts along the number line.

Most actual arithmetic is performed either as modular arithmetic or by a kind of mental gear ratio twisting: because I remember all the ways in which integers can be combined to reach 10, and because I have a strong mental image of counters ticking up and down, I can see how addends and subtrahends 'wrap around' and overflow or underflow at the radix. I do not actually have most of my times table completely memorized by rote for all combinations of single-digit numbers, but small number multiplication is simple enough to rebuild out of geometric intuitions on the spot, and the large numbers that should be problematic are trivial because I can visualize them as ratios of addition and subtraction. (e.g. to increment by eight, add one to the tens place, and subtract two from the ones place, because eight is two less than ten.) (an analogous line of reasoning also demonstrates why all the products of nine between nine and eighty-one have digits which themselves add up to nine.)

I have few problems with exponentiation and radicals, which can usually be reduced to smaller addititons, subtractions, multiplications, and divisions. Conversely, my representational schema and mental machinery for long division and logarithms have atrophied greatly between now and when I last took a fundamental math class, not helped by my being able to lean on electronic calculators at all times.