Thread

On Math Hype

This thread is to show that math is overhyped. Of course, such a hot take requires a disclaimer, so here it is.

Disclaimer. According to Mamardashvili and Pyatigorsky's Symbol and Consciousness, a symbol is something that produces a complex concept in your mind that is not contained in the bare object underlying the symbol. For instance, a dove is a symbol of peace—a complex concept—which is by no means contained in the bird itself, but is somehow sparked in one's mind upon seeing a dove or its image. In this way, a "symbol" consists of three components: the raw object that symbolizes, the complex concept that is symbolized, and the link between the two.

The tendency with symbols is that most people are interested in the symbolized concept rather than the raw object underlying it: people dispute about peace, whereas doves are only of interest to a small group of ornithologists.

The same is true about math. There is the "raw object" of math—such as all the math papers in the world—which is only of interest to a small group of mathematicians who pursue math as their main occupation in life. Then, there is the complex concept that is sparked in the minds of laypeople when they hear the word "math."

This post is a critique of the latter.

So, what does "math" symbolize? On the surface, the answer is "humanity's understanding of the world." On a deeper level, this transforms into "humanity's control over the world." Here is what Jean Baudrillard writes regarding one of "math's" triumphs—the moon landing [1]:

"The exaltation of the crowds was not a response to the event of landing on the moon or of sending a man into space [...], rather, we are dumbfounded by the perfection of the programming and the technical manipulation, by the immanent wonder of the programmed unfolding of events. Fascination with the maximal norm and the mastery of probability. Vertigo of the model, which unites with the model of death, but without fear or drive."

Indeed, it is unsettling to live in a world where you can be struck by Zeus's lightning at any moment. It is much more comforting to live in a world where lightning is electricity and electricity is under our full control—we even have a formula for it!

Thus, we inevitably arrive at the opposition between "math" (or "science"—these two are equivalent as symbolized concepts) and "religion." This opposition is absurd at the level of raw objects: there is no math paper disproving the existence of God, nor can there ever be. However, the clash is truly epic at the level of symbolized concepts.

i-1-24 (A Soviet drawing with a rhyming caption: "We've searched the sky throughout—there are no gods or angels.")

What is "math's" position in this clash? That it provides the "true" model of the world. But this is a cruel, subtle, and misleading twist. A model cannot be assigned a truth value; rather, it can only serve your purpose better or worse. This means model quality is fundamentally purpose-dependent. By lobbying for "math", one actually lobbies for a specific system of purposes while camouflaging their intentions as a search for truth.

What is the purpose behind "math"? As discussed, one purpose is to feel in control of the world. Imagine two people sitting on a bench contemplating a rainbow. One thinks what they see is light being dispersed (refracted, polarized, or disintegrated—pick whichever word makes you feel smart) by an aqueous suspension. If they wish, they can reproduce the phenomenon with a prism, just as smart Isaac Newton did. The other observer is an adept of an Abrahamic religion, and to them, the rainbow is a symbol of God's covenant with Noah—a symbol of divine mercy. If they wish, they can strive to purge themselves of sin, just as righteous Noah did. As you can see, this is not about true versus false, but rather about controlling the world versus dwelling in supreme mercy.

Religion is an obvious example of a domain where "math" has no foothold, though it is far from the only one. Yet, "math" harbors the ambition of becoming omnipotent—of entirely encapsulating the world. "I want to know God's thoughts; the rest are details," reads a quote attributed to Einstein (who is himself a symbol of science, and thus a derivative symbol).

The impossibility of total omnipotence, however, forces "math" to commit a murder: shifting from describing the world to defining it. "If you cannot measure it, you cannot improve it," a sentence attributed to Lord Kelvin, a British physicist. What a bullshit, your Lordship! How about a musician practicing a piano piece? If improvement only lies within measurable domains, then the quality of a musical performance would be solely defined by rhythmic precision. How much richness is sacrificed here just so music can be intelligible to "math"!

Or take a basketball game. If I ask you to predict the outcome, what will you think about? Who will win, presumably. Why? Why not think about who will display more dedication on the court? That is because "math" has almost monopolized the act of predicting the future (we mock the very concept of an oracle), but "math" can only operate with numbers. Dedication cannot be quantified, whereas the final score is readily numeric. Thus, the result of the match becomes synonymous with the final score—or perhaps individual player statistics if you want to dive deeper. A perfectly intelligible, predictable desert.

All in all, why is this discussion important? Why don't we just reduce the world to what is countable (or uncountable, but strictly in the set-theoretic mathematical sense!) and live in the comfort of total control?

Because we seek more. We seek purpose, meaning, beauty, and mystery. If Bob's job is to add and compare numbers to maximize other numbers, he can do it precisely and efficiently today—but he will eventually burn out from the lack of underlying meaning. "Math" will not help Bob here. It will claim that meaning is not well-defined and therefore cannot be maximized. It is Bob's responsibility to keep "math" bounded within the domains to which its raw object belongs, while cultivating himself in other domains, mastering orthogonal knowledge.

math philosophy semiotics

4 replies

Sources1

ArticleSimulacra and SimulationWikipedia
en.wikipedia.org ↗·Cited by 1 post

Replies

Henry D’Angelo@henry·Sep 16·edited Sep 16
This is beautiful. I will respond to one specific thought --- the claimed target of math being omnipotence.

To set the conceptual stage for my response, let me first appeal briefly to some notes.

Question: How do you quantify randomness or "patternness"? One of my favorite proposals is the concept of Kolmogorov complexity, which quantifies the randomness of strings relative to a computer.

Informally, the Kolmogorov complexity K(x) for a string x is equal to the length of the shortest computer program that can output x. So, what does this mean?

Consider the ζ (which is just “10” repeated 50 times):

ζ: 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010

This string is 100 characters long. But its Kolmogorov complexity, K(ζ), seems to be less than 100. Why? Because you could (in a given language) program ζ in less than 100 characters. I essentially just did; in pseudocode, it would be something like:

print(“10” * 50)

This program, despite being shorter than ζ itself, would output the string correctly. So you get the idea: the Kolmogorov complexity of an object in effect reflects how long it takes to be described (though the specific measure varies across different languages and computers, the idea is consistent across).

The basic thing to note is this: there is a sense in which ζ seems not to be random; intuitively, it is the fact that it has a pattern. And it is this pattern—repeating “10”—that enables us to describe ζ in a compressed way (that is, using a program that takes less characters than the string itself). Kolmogorov complexity is meant to formalize this intuition — that fundamentally, patterns enable compressibility, and randomness is incompressibility.

So, Kolmogorov complexity (or algorithmic information) captures the intuition of patterns being dual to randomness, which is neat. That said, it's not generally computable, and there are some technical things needed to make it work formally (e.g. the idea of prefix-free Kolmogorov complexity).

Now, how does algorithmic information relate to the question of math's potential for omnipotence?

Informally, I generally think of math as follows: "the methods of compression, universal across effective compressers, in the limit of data."

So, what is an "effective compresser"? Roughly, imagine there is an algorithm that receives datapoints (in the "real world", think sense-experience), and then performs some kind of "operation" to encode them. There is time to try different schemes. Loose example: a scientist who is reasonably intelligent. The scientist observes the world. He seeks a "theory" to understand these datapoints, and he has his lifetime to do so. His basic goal is to find "programs" for compression that allow him to parsimoniously "recreate" these datapoints, which in turn, helps him to predict them (assuming his methods of compression identify true "hidden causes", and ignoring the problem of induction).

Eventually the scientist will arrive on certain programs for compression. Maybe he'll find numbers. In fact, with enough time, he is sure to discover "something like" numbers. So perhaps there are indeed methods of compression that are universal or "inevitable" for effective compressers with enough datapoints. This is what I imagine math aspires to.

For example, maybe addition is one of those universal methods of compression. What I'm saying here is that if you're trying to compress datapoints, and you have enough datapoints, eventually "something like" addition (up to isomorphism?) will be strictly useful to encode. Maybe. And, so any compresser that's good enough and has datapoints, will do so. That is to say, perhaps any scientist that is sufficiently advanced and has enough time / data, will eventually discover addition.

Quantity, length, etc. these things strike me as being potential "universals" in this above sense.

But then...

Are there such universals? Does this concept even make sense?

A problem is that Kolmogorov complexity is computer/language dependent. You can force it to be a universal computer, sure. But the reality is that relative to any computer, even a universal one, the vast majority of strings are random (I can prove this fact quite easily, if you'd like -- the basic idea is that if you define "not random" as "described with a short program", you run into the problem that there are many more long strings than short strings, so short inputs can cover only a small part of the terrain) then there is always a computer to which those "random" strings aren't random.

The extent to which patterns or randomness are necessarily observer dependent is a genuine facet of math being limited. Even in the formal world, compression is limited.

Further yet, there is a deeper point, which perhaps is related to what you are getting at. The point is this: it is not just the question of whether "datapoints" (left undefined) are patterned (compressible) or not, BUT the fact that it is a kind of fluid act or "process" of actively receiving datapoints, being affected by them, making sense of them, etc. that is itself a totally different thing. You can never "complete" this task.

You don't get anything whatsoever about firsthand experience from "universals"; in fact, firsthand experience might be described as "that which is not universal".

A scientist who observes the world and discovers some beautiful pattern or order; this is a beautiful thing. Is there any way to "compress" this experience? Maybe not. Some things are defined by their limitations. Can anything be defined from the outside? For that matter, can anything be defined from the inside?

Fixing things in formal terms delimits those things, but delimit the world it does not.

I am still in synthesis mode, but these are the sparks that were struck for me. Hopefully some are enjoyable.
Anton Paramonov@anton144·Sep 16
Not that serious of a comment, but related. So there is this code-golfing enterprise, which is about solving a programming problem using as short of a program as possible. For instance, consider the problem of taking a body of text, splitting it into individual words, counting how many times each word appears, and outputting the words sorted by their frequency. In a standard Python way, it can be solved like this (details not important):

import re from collections import Counter def count_words(file_path): with open(file_path, 'r', encoding='utf-8') as file: text = file.read().lower() words = re.findall(r'\b\w+\b', text) word_counts = Counter(words) return word_counts.most_common() frequencies = count_words("f.txt") for word, count in frequencies: print(f"{word}: {count}")

but the golfing solution will only consist of 114 bytes:

from collections import * print(Counter(open("f.txt").read().lower().split()).most_common())

However, if you use another programming language, e.g., bash, you can get away with just 19 bytes: xargs -n1<f|sort|uniq -c.

So code golfers realized they can beat records by switching to other languages. This eventually culminated in them creating new languages and, finally, and this is the punchline, they came up with an idea of enumerating all possible programming languages, so that the solution to the problem above in programming language number 41064167836102135290461892 takes 0 bytes and looks like this: .

I'm not an expert in Kolmogorov's theory, but it looks like those nerds outplayed him somewhat.
Anton Paramonov@anton144·Sep 16
On a more serious note, what you are writing has this Kantian twist that when we are exploring these universals, we are exploring our thinking rather than the universe, "what helps us compress the universe" instead of "what helps us compress the universe". Under such a twist, one wishes that our thinking is a constant object, or possesses some constant principles, so that it can be studied. That's kinda an implicit axiom of Kant that somewhat fired back: he was assuming that our perception of time and space is constant and does not change with experience, which contradicts the very phenomena of relativity theory that did exactly that: changed our perception of time and space based on our experiments. I believe you also have this axiom if you are striving to extract some universals.
Henry D’Angelo@henry·Sep 16·edited Sep 16
P.S. this reminds me of how we discussed the definition of a game. You suggested something like: a set of rules with a goal attached. I said something like: a set of constraints that an agent adopts to further their own goals. I believe when it comes to games, just as it does with signs, you need to highlight the thing "outside" the game (the "concept"), not just the game itself (the "object" or sign). Ultimately, I think what matters most is the "bridge" -- from within the game itself, to the player who is "outside" the game.