I've always liked the idea of math as a type of language. When you learn quantum computing, for example, you go beyond the limits of our informal language (English, etc.) which developed in a purely classical world. This is why pop science articles on quantum phenomena never really make any sense. They try to use metaphors to explain concepts, but there is no existing concept in our informal language to use as a referent. Many quantum phenomena belong to an ontological category that is entirely new to most adults. For example, superposition: you try to explain it as something being in two places at once, but this doesn't make sense nor is it very helpful for understanding how objects in superposition behave. In order to understand quantum concepts you have to learn a new language - that of mathematics. Then it becomes easy! Superposition is just linear combination. Done.
Also if you enjoy thinking about this topic I recommend Neal Stephenson's book Anathem, which I just finished. It is a complete slog at the start, with endless boring architectural descriptions. I listened to it in audiobook form which made that section much more pleasant.
Math isn't a language, it has a language. You can easily invent your own language to describe different aspects of maths. Saying that it is a language is like saying that computer science is a language since people write C etc.
Consider an EM field. Do you think there is some daemon grinding out complex exponentials as it propagates? In reality some spinny thing is rotating and moving through space. What those terms actually are isn't precisely known but the phenomenon closely fits a mathematical model we can elaborate on to do useful things. That doesn't mean it's how reality works.
Though it could be how reality works. Maths things like pi don't seem to need daemons calculating it for it to have a value. And perhaps likewise EM fields.
It has been demonstrated that humans have no easy way to universally distinguish artifacts of perception from empirically real phenomena. So I really take issue with this use of "exist". It's never that simple.
And that's before getting into categorization issues. Does Pluto "exist" as a planet? It'd be pretty non-sensical to say it stopped existing when scientific authorities changed its designation. But then was Pluto really a planet before that happened? Or was its planetary status a figment of collective imagination?
Are you suggesting that the concept of symmetry may conceivably be an artifact of perception? Though I accept the statement of your first sentence, I find this application of it extremely hard to accept.
I agree categorization is a (almost?) purely human concept. I'm not sure how that relates to my question.
Yes. This extends not just to things like symmetry but also to what we think the concepts of "space" and "time" to be. The Critique of Pure Reason is the classic work on this.[1] There Kant defines the phenomenon and the noumenon, sort of as aspects of everything that isn't us. The phenomenon is what we perceive, the noumenon is the "thing-in-itself" which lies beyond our perception. Though there's an open question if it's beyond our conception, Kant left that unanswered, which is where the ensuing debate over the nature of math comes in.
Why does this relate to your question? Because we cannot prove _anything_ not to be an artifact of our perception, nothing that we conceptualize is necessarily inherent in reality. In a strict sense, reality is unknowable to us. In a practical sense, none of this matters because math seems to work well. Still, if we wish to be undogmatic, then we should approach questions of existence or non-existence with humility and not base ontology entirely on perceptual illusions.
When you do math you invent new language, ie you add notation and describe what they mean. The math is the actual concepts, not the language you create, an other person could have structured his language for this problem very differently yet ultimately describing the same thing.
Edit: To clarify better, the way you solve math is that first you solve the problem, then you figure out how to communicate the solution. This can either use language people created before or you create new language, you can just use English or you can just draw pictures and have no words at all. Either way you are doing maths.
I'd clarify that when people talk about "a different language", they're usually describing the notation used in formal mathematics. But that's a comparatively recent phenomena (most of it from the last century). For most of its history, mathematics was (and still is, most mathematics papers have far fewer uses of the notation than most laypeople would think) communicated with the natural language of its authors and their readers. The problem was that natural language was frequently too imprecise to easily describe things formally (inclusive vs exclusive or is an example), so mathematics writing became stilted and full of jargon and structural formalisms: "consider two values that are natural numbers and neither of which divides the other without remainder and ..." versus "$$ \{x,y \mid x,y \in \mathbb{Z}_+ \ \mathbb{and}\ x \perp y \}$$" (Edit: wait, Hacker News doesn't support MathJax?) Modern mathematics notation is more to simplify clear communication rather than to describe things that can't be described with words. That is, mathematical notation has far more in common with shorthand than it has with Esperanto, and I think that's an important distinction. When I read set-builder notation, what I "hear" in my head is English. (Also, you're begging the question by implying that there is an external concept/thing that the various languages and pictures are referencing)
When people talk about different language, they mean more than just notation. For example, I can write "2 | 4" as "4 = 0 in Z/2Z". There is more than just a notational difference here.
A more non-trivial example: Say you want to say that 2 x 4 = 8. One way is to just write 2 x 4 = 8. Another way is to say that a finite set of cardinality 8 is the product of a finite set of cardinality 2 and cardinality 4 in the category of finite sets (in the sense that 8 satisfies the universal property of 2 x 4).
How is your example different from claiming that the statements "Aaron and Elizabeth" and "My friends whose names begin with vowels" are statements in different languages? I also note that with the exception of the symbols for multiply and equality, your final statement is written entirely in (an admittedly stilted and formal dialect of) English. I'm not saying that Mathematics is anything but beautiful and expressive---but the only way to justify a declaration that it's a distinct language is to do irreparable harm to the meaning of the word "language". And, of course, you make my point in your first sentence. I agree that what we call language is more than just notation. Which is one of several reasons (among them concepts related to things like mutual intelligibility, orthography vs. grammar, native speakers, effortless learning during early childhood, communication as an evolutionarily selectable trait, ...) that I disagree that mathematical statements or reasoning are the same species of thing as British Received English, American Sign Language or even Elvish and Klingon.
Well, if we're allowed to just redefine the terms, you can make and prove whatever claim you wish, but its unsatisfying and ultimately pointless. I'm not sure why you think the above (which, I'm sorry, parses as gibberish for me) is useful given my assertion that mathematics is not the same species of thing as human language, as opposed to whatever the thing you're describing above is.
You don't really "solve" math. You apply math to a concrete model. For example, if you want to build a four-sided pyramid and all you have is a really big straight edge and a really big collapsible compass, you'd probably want to read Euclid first. And if you happen to be doing this 2000 years before Euclid you'd read some of the geometric methods his work was based on. To the extent the surface you're building on is "flat" the axioms and the theorems apply. And so you can square a circle easily using simple tools like cord and stakes. Thus you can build a pyramid with some cordage and a stake.
Doesn't this still depend on what specifically "math" is referring to? Sure there is some underlying concept, but when people use the word math I think they are largely referring to the language that has been/continues to be built up to describe those concepts. Without at least learning the existing language one would have to rebuild a ton of foundations themselves before they could do much of anything. I suppose you could say the same thing about e.g. biology but I don't think it's quite analogous, as terminology is nowhere near as precise and things often don't abstract super nicely.
E.g. the word “carbohydrate”, or it’s equivalent in different languages, describes a concept; but the concept itself exists independently of any language construct that describes it. Similarly, in math, a concept e.g. “prime number” is independent of the many ways and many languages you can use to describe it. It just is.
It's less clear in informal languages that concepts exist apart from the language to describe them. This is one of the great difficulties in translation; different languages conceptualize the world in different (and sometimes incompatible) ways. For example, Latin has a variety of words for relationships of power that don't convey cleanly into English. The Greek word λόγος also doesn't cleanly render in English. Similarly for much of the Tao Te Ching. To what extent our language shapes our conceptualization of the world is currently an open question, but it seems like it's at least some.
These seem like extreme cases that just take more words to translate due to limited overlap of shared experiences. If it never snows in Hawaii, the Hawaiian language may not have a word for it. But snow is a real thing apart from language. I think what is meant by informal language is more the structural rules, which ultimately seem to just be arbitrary patterns of behavior people fell into, instead of sometime deeper.
Those rules too exhibit cross-language conceptual shear. For example, English doesn't really have an equivalent of Latin's subjunctive mood, and our clause boundary system is way weaker. We use punctuation marks in a way that the Romans didn't (they used punctuation words in a much more systematic way than we do). Greek has a separate grammatical number for pairs of things as opposed to our singular and plural. Those rules, in addition to the individual words (perhaps) bespeak a different way of conceptualizing the world.
It actually does snow in Hawaii. People on the big island have skis and snowboards for the few times it snows on top of their mountains. It also snows on Haleakala in Maui which is pretty fun to think about.
The concept of "a number that cannot be written as a product of two other numbers" just is. It might happen in some setting that none exist, but the general concept most certainly just is (and they can be shown to exist among the natural numbers if you insist - but they do not exist in e.g. Z_2 - the binary field).
Even in your own comment, equating Superposition and linear combination, you are almost there but not quite. A spanning set of linear combinations is not the same thing as reference to the concept of linear combinations. One might argue small difference, but we are talking about the precision of language here.
In my understanding, that’s exactly what math is, a language. And various modes of thought have utilized the language of mathematics to communicate ideas that were otherwise not possible or not consistently able to arrive at same conclusions.
This is very interesting. What is language? When does language evolve? Is language itself consciousness?
Whenever there’s a leap in language is it because something had to be constructed to help assign an abstraction to a coordinate in space where thoughts exist?
It appears we’re languages all the way down.
Or analogicalmachines? Lol.
Lots to unpack here - some with what many would call not nice Consequences.
What happens when a minority elite (unsure how else to describe it?) disciplines require a language that the majority lack the finesse to use? Example: math used in physics, or programming in computation. If the languages being used introduce more complexity without the ability to reduce net complexity then it damages the communication of the specie using it. I’ve been thinking about this quite a bit on our trajectory as a species.
Complexity in language leads to information silos which leads to various existential risks based on information and miscommunication.
Also interesting coercion / authoritarian emergence comes out of those creating a new language to reason about an abstraction which leads to a success that negatively impacts those which do not know the language and if the space to understand said language is sufficiently far enough you have much room for manipulation. Also, who gets to say which language is the right one? What happens when a new language is created that is more advanced and the majority disallow it preventing an evolution? Interesting dynamics.
We don't have to stumble around blind here. Philosophy has long pondered the nature of language. A more recent idea is that a language as we think of it - English, French, etc. - does not exist per se; rather we engage in "games" with each other which are languages in themselves. For example, when I buy something from a shop I am playing the "purchase an item at the cashier stand" game. Both the cashier and myself know all the valid moves in the game - they say the price, I provide a method of payment, etc. Some of these moves involve use of a limited subset of English words, but even if I'm in a foreign country where the cashier doesn't speak English we can usually make it work. This interaction is a language in itself. If I were to walk up to the cashier and try to play the "negotiate salary for a full-time position" game, only confusion would result - I would be speaking the wrong language, even though I'm emitting sounds which we both find familiar. Much confusion and anguish in the world results from people playing the wrong language game with each other - for example, one person playing the "venting frustrations" game while another plays the "provide advice" game.
In this analysis, the "explain quantum phenomena" game is just a language where the rules are especially strange and prone to pitfalls. Deploying vocabulary from mathematics, a system with very well-defined and mutually-understood rules, enables us to navigate this space.
Yes, I see humanity engaging in games and LARPing as an interaction between those who are aware of the game being defined by arbitrary rules, and those who don’t. Most don’t realize it’s arbitrary.
I’ve thought about what you said recently and it comes down to “well-defined” being relative. Usually to those that are defining it. It’s not objective. Additionally that’s enforcing or coercing a game unto others that they may not wish to engage with or may not be able to conceptualize it.
Some people prefer characteristics over others. Have predispositions to likings and not likings.
Using a language is in of itself agreeing to a game.
If a thing requires consent of will to proceed as a rule, it’s a game.
Language, or any system of rules would be part of agreeing to a confined space of possibilities and “game-space”
Interesting to ponder nonetheless at a different abstraction. I keep trying to go “deeper” behind the language itself, I arrive at signal and then we’re in a puddle of discrete and non discrete values. Fun.
Growing up my father would often describe a successful person as someone who "knew the lingo" of whatever they were successful at. As a smartass teenager I thought he was oversimplifying things and discounting other factors which didn't involve, what sounded to me like, being a BS artist.
An alternative definition is that language is a set of referents to a collectively constructed "picture" of reality. Wilfrid Sellars has done work on this.
I say alternative because it shifts the focus away from person-to-person interaction and more towards impersonal social processes that construct what we think of as "objectivity". That's not to say what you're referring to isn't also an important part of the picture so to speak.
Mathematics (and logical reasoning in general) is almost impossible without a language, but in and of itself mathematics is not and cannot be reduced to being just a language. A language is invented, math is discovered. On the other hand, in science (and inside mathematics itself) the word “language” is often used in a relative sense, in which even chemistry, for example, can be called “a language.”
Folks working in Foundations of Mathematics would like a word...
On a serious note even if objective mathematical entities exist, which I personally don't believe is verifiable, it'd be really something to see how they can maintain said existence while satisfying Gödel's proven theorems.
Your assertion that math is discovered is just an axiomatic statement. Your argument relies on it as bedrock. If I don’t except that axiom and prefer to start from math being invented by conscious minds then we can never get anywhere. I think the entire ‘meaning of mathematics rests on assuming one of these axioms. I can not accept a claim that mathematics is discovered, therefore, calling math (which I personally view as a logic) a language is perfectly valid.
> I can not accept a claim that mathematics is discovered
Why?
Finding out that Pi is irrational and then transcendental are discoveries. I won't be able to make anything work if I fork mathematics to make it rational.
Same thing that finding out that if you tie a string to a stake and then go around keeping it tight, you have to walk a distance that is two times pi times the length of the string to get where you started. I don't need a language for that, just more pieces of string.
You might be interested in an influential essay called "Higher-order truths about chmess" by Dan Dennett. Chmess is a hypothetical variant of chess with slightly altered rules. There are infinite interesting theorems you can prove about chmess. But what is the value of doing so? Are these theorems discoveries?
Pi is not calculated by running around in a circle with a string (nor by any other real-world measurement), it's calculated by setting up a rule-based system (math) and following the rules to derive its value. You can get pretty far in the real world by using a rational value for pi - ten decimal places should suffice for atomic-level precision, I believe.
One of the interesting consequences of the recent redefinition of the SI units is that now h, k, e and a few others are considered exact constants, just like c. A caveat: physicists are well aware that their trade is about mathematical models of magnitudes they can measure (and extra stuff they can't yet comes in handy to do that), not reality or the physical world in itself, whatever that might be it's something dealt with epoché.
So now the Stefan-Boltzmann constant, which involves powers of c, k and h is exact. It also involves the fifth power of Pi (through the value of the Riemann zeta fuction for argument 4). As you can measure the Stefan-Boltzmann constant being creative with a radiometer, you could determine increasingly accurate values of Pi experimentally. In the sense of magnitudes as used in physics, which is a fruitful approach to being quantitative in this confusing physical world, this "thing" we call "Pi" behaves like a dimensionless physical constant.
Before you'd object, if we came up with more precise theories, experience shows that that would mean corrections to the laws implied above. It also shows that when this happens instances of "Pi" don't deviate from the Platonic value of Pi.
“ physicists are well aware that their trade is about mathematical models of magnitudes they can measure (and extra stuff they can't yet comes in handy to do that), not reality or the physical world in itself”
This is entirely oxymoronic and double thinking and exactly the type of mental gymnastics that manipulates what truth is. Physics is based on the physical because it was built upon observation of reality using our senses. Yet modern physics is based on the imaginative and has nothing to do with the physical.
Any measured unit is imaginary. The only thing that exists is the measure.
Exists is based on physical. To be is to exist. To exist must be within physical reality. Without this distinction then we might as well give into the esoteric notion that everything is entirely a mental construct and nothing actually “is.”
How far we’ve come from the physical to the mental and then lack the nuance to distinguish as such and furthermore blur the lines is a disgrace to critical thinking and cognition.
Physics isn't ontology. You perceive a phenomenon, pick quantifiable observables, take measurements of them using units which are nothing more than reference measurements, come up with a mathematical model for those observables and test its predictions. That happens to be the best you can do with inductive reasoning. Believe it or not we don't establish reality, we just live in it.
This is only a general comment because I don't understand anything you've written so I can't address your points, which undoubtedly must exist.
Pi is a concept that is divorced from reality, Pi is irrational because a chain of logical inferences and applications result in the definition. The definition of Pi is predicated on a geometric description of a circle which is distinctly non-physical. The entire meaning of Pi is based on a logical chain of defined symbols, that while meaningful to some conscious thinker are not observations of physical reality. The geometric conception of a circle as an infinite collection of points, which are inherently not physically possible objects, exhibits a ratio between its radius and its diameter, which is called Pi, and that ratio needs to be expressed by a number which is definitionally irrational (the definition of which is predicated on a definition of Natural numbers (via a definition of the Real numbers)). These symbols are merely the language we use to describe logical relations between various symbols, that while possibly applicable, with prediction and accuracy preconditions, to physical reality, are merely a logic used to describe natural numbers and the properties thereof.
It is this complete logical decoupling from any physical, observable reality, that I cannot accept mathematics as anything other than an, ideally, internally consistent, logic used to define and discuss symbolic logical objects. And I am of the opinion that logics are merely invented by conscious beings as means to systematize areas of discourse and to derive internally consistent inferences and implications built there on.
I would agree that if irrational numbers were prohibited from being inferred in the logical system of math then some things would not work within the logic. Because a change in the axiomatic or inferential rules of a logic are changed the logic as a whole changes. But, if, as you describe it, math was forked to be limited to rationals, and the precision of those radicals was calibrated to be on the order of 1/10 of a Planck length, I am not convinced that any equation attempting to describe 3 dimensional space would be inaccurate. However, I could wrong.
It’s similar in my perspective to a nuance that exists pervasively today within scientific models. Just because the model allows a prediction doesn’t mean that the model reflects “base reality.”
Our models, our tools, are the results of our observations. If it works to provide a prediction, then it becomes useful. However I feel there’s a disconnect as the nuance of a model vs what is, has been lost among many.
The map is not the territory.
It’s very interesting to see science use mathematical models as objective reality and orthodoxy. I’d say that isn’t science, but words change in meaning with the zeitgeist. Unfortunate in many ways.
I like the way you framed the argument.
Any recent things you’ve read in this realm of subjects that were interesting to you? Any favorite past time papers/books you’d recommend?
I certainly have a lot of recommendations that are topic adjacent, but most of my comment comes from mental revision. I have a horrible habit of reliving previous arguments and trying to find out what I could have said better to win the argument. I had an argument with a law school classmate who had a Masters in Mathematics (now a PhD) while we were on a road trip from Paris to Normandy, during summer abroad, and then ‘invented/discovered’ nature of mathematics was the real crux of the argument. I have spend the last 12 years going over and over and over that argument trying to win it and my comment is a result of that. So my recommendations are not going to be super on point, except to say that the Stanford Encyclopedia of Philosophy entry on the ‘The Philosophy of Mathematics’ [1] and the large list of citations are great, particularly the ones from the sections on Formalism and Fictionalism (but these are not the most interesting reads if you’re not digging for a mic drop quote for your imaginary debate).
For some topic adjacent past time papers, a lot of my comments concerning logic come from the research for my programming language, so I have been immersed in logic and proof theory work for the last six months pretty hard. I think any of the course note PDFs from Frank Pfenning (CS prof at Carnegie Mellon Univ.) are great read in general (and are easily found by googling Pfenning logic course notes and just looking around). If you like video lectures, or just listening to them, any of Pfenning’s lecture sets from the OPLSS session, which are all on YouTube are wonderful, particularly the 5 lectures from 2017 on Substructural Type Systems and Concurrent Programming [2] Also, Noam Zeilberger’s OPLSS lectures on Refinement Types [3] are great and his dissertation was a great read [4]. Finally, Neel Krishnaswami and Dunfield‘s 2 papers on Higher Rank Bidirectional Type Checking [5] was really good (and like all the above the cite lists are a trove of good stuff).
I literally could keep going for hours on great reads in logics and type theory, so if you want more in some area I’ll provide.
Not very divorced. I can glance at my coffee cup and notice that there is a distance around the circular edge and another if you were to go straight across. Maybe they are just having a trial separation?
Pi, or the ratio between a geometrically defined circle’s circumference and radius ONLY holds as a truth in so much as one is discussing geometric circles. Pi does not describe the physical world, it describes a relationship between logical objects.
But it is uncommon in my experience to actually define Pi as this ratio, if you build math axiomatically; Rather, the "shortest way" to get to pi which is well defined is by first defining the exponential function ( \exp x = \sum_i=0^\inf \frac{x^i}{\fact i} ) with all the pre-requisite for that (numbers with order, addition and multiplication; limits and convergence; then imaginary numbers). Then you define pi to be the smallest positive number such that exp(2pii) is 1, and e to be exp(1). All the properties of pi follow "easily", including it being the ratio of am euclidean circle's circumference to a diameter.
The thing is, in math, all of these things end up the same regardless of where you start; Whether you start with a geometric definition of a circle and work hard to discover said ratio, or you start with exp(), you'll end up with pi=3.14159.... and it having the other properties. It is in that sense, not arbitrary.
You could (and would) take a step back, and say that being euclidean is arbitrary - which is true; but any description compatible with euclidean axioms will get the same value of pi as the ratio of circumference to diameter; and that value will be the same of the exp() value that has no concept whatsoever of euclidean space.
It is in that sense that math is "discovered" - there is no euclidean construction in which pi is different. There is no peano-compatible (set, games, p-adic, or other) construction of natural numbers in which primes do not exist. The peano axioms themselves are, indeed, logically arbitrary. But the "discovery" is that "peano -> existance of prime numbers" -- and that does not depend on language.
I think ratios are closer to existing as something real as the modification of a ratio has physical implications. Ratios are discovered, math is invented.
Your statement of disagreement is also an axiomatic statement. The laws of physics tell us that aliens that think would discover Pi, ergo, not a language in the normal sense of the word.
I conceded that by stating that both were axiomatic positions. I also feel my position is more justifiable. Language in this case is a formal system of using and manipulating symbols to enable implication. If the ‘laws of physics tell us’ some hypothetical alien being ‘would discover’ Pi, that just means said aliens had to discover a logically consistent meaning of the natural numbers, and I would contend any species sufficiently abstractly intelligent enough to do so would also ‘discover’ Boolean logic. The fact that some species can reason using implication on the Natural numbers or some isomorphic inductive structure does not mean that the logic which enable those implications is anything other than the product of the mind, i.e. an invented thing.
If following the logical inference chain that leads to determining the geometric concept of a circle and then inferring the ratio between the circumference and radius is a discovered thing, then any and all logical inferences which can be proved are then discoveries, but I don’t think that argument can be taken seriously.
It really is worth it if you get past that part, which I completely agree is a total slog. The audiobook is very well done. Things pick up considerably once you get to Apert.
It’s a great question. That depends on if an emotion and image would be language.
Given we had no language and communicated telepathically I’ve imagined that:
1. Using a feeling
2. Using a mental image
May be enough to form abstractions but that may in of itself be within the set of languages.
Said another way though, does language require symbols? It’s certainly possible to imagine language without words and we live in a time where symbols are abstractions for entire concepts within the noosphere.
Language is fundamentally based on abstractions.
Abstractions are fundamentally based on mind.
If the mind conjures abstractions then is it fundamentally a language no matter what? Probably.
What are the ways that we may use language to communicate? Far more than we think of today. Every time we extend the abstraction we are essentially evolving as a species. Almost to the point where our evolution is entirely dependent on our ability to abstract novelty continuously.
There are many who are unable to form images in their mind. Aphantasia isn't rare. But it's also not universal and people with aphantasia somehow manage to communicate to those without well enough most of the time. Perhaps then language isn't just visual images or purely mathematical abstractions.
I think the view of Wilfrid Sellars, basically language as a collectively-authored set of referents to a non-visual conceptual "picture" of reality, matches well with what you're saying here.
Yeah I just came across some similar ideas with reality being a shared hallucination that is based on aggregate consensus. Similarly the universe is an eventually consistent algorithm to allow that private state of each human to converge on a shared reality.
It's clear that every animal has the abstraction "food", which applies to groups of molecules and atoms, and does not require anything like a language.
Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all?
How does one find ethical or religious problems and questions in pure logic? Logic is fundamental and proven to be able to apply in reality. Ethics and religion operate over multiple levels of qualitative transitions and cannot(?) be fundamental.
Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)
Because DoJC exists outside of space and time in minds of believers? It just doesn't have a physical counterpart and does barely have any logic to evolve/deduct from it, for us to predict events based on DoJC as if it was applicable to parts of reality. It is understandable how someone could ask "stupid" (not really) questions about maths, but when philosophers start to draw parallels, it is easy to lose track of what they are even talking about. Can someone explain what they meant exactly?
There are some core questions in ethics that might be considered so fundamental that they almost don't seem like religious question at all.
The Observer Problem is a good example. I experience life from this singular human's perspective. Why this specific one? What mechanism dictates which human my "observer" occupies? Why even a human and not an animal?
The word "soul" is a poor descriptor because it is associated with so much more, but the Observer Problem is real - or at least it might be.
I heard a theory is that there is another higher order time dimension, and all of our "observers" are actually the same one living in some sequence "simultaneously". ...but that doesnt really solve the dilemma either because even that sequence would need some mysterious ordering mechanism.
...but in all that there are core questions about what is time, what is space, what is energy, what happens to the universe... etc
One thing I tell me daughters is that death isn't scary. ...because we exist inna limited space and time. We should be no sadder to not exist in the future than not existing 10m to the left. We cannot be eternal, but if you recognize that time is just a dimension, then our existence is permanant in the history of the universe. Nothing can undo what we did and who we were.
Sorry for off-topic, but The Observer is my great interest as well. Why not an animal? Sure, why not. Imagine that your ears, eyes, taste, etc are shut down and your consciousness too. Now you feel yourself as a rock. Why not a star? Why not a computer game character? There is a bunch of problems to think of before you answer "no". The same for "yes", because there is a theory that it's a human mind that thinks of that Observer concept and makes itself "feel" it, but... it is not real, a perception mistake required for a mind to function in reality, like many others.
This has something to do with models, but it is hard topic on its own. It seems that The Observer is one of things that don't have a quick answer and may not have one at all.
With distinct Observers (like distinct photons or distinct fires) you have to solve the "where it comes from and where it goes" problem. It is not a problem if you think of it as an Observing Field (like em-field or plasma state). There is no "why not", because sure it is everywhere, but only complex enough interactive systems can ignite it at their current location. That's still pure speculation, but more aligned with Occam's and has to be considered as both a competing idea and as an answer to "why in me and not in xyz".
I stopped at this crossroad:
- distinct and only me (1)
- distinct and multiple (N)
- a field-like phenomenon (~inf)
- a perception mistake (0)
First two bring a hierarchy problem (why, how). A third has a scientific potential-like flavor. The last one is boring. Not sure if I ever get any further than that, sadly.
That's rather a simple one, every point of experience is "this specific one" for itself, but in reality there are no specific points. If you were a fish in previous life you would only know that and the question would be the same.
If you mean "why" in terms of "how", the mechanics of it, well as far as I can see it's way bigger than our intellect and in turn language can handle. One assumption I have is that at the moment of death we will get it, not intellectually because the brain won't work but rather experientially.
>our existence is permanant in the history of the universe
Tell me how do you really, existentially separate yourself from the universe?
A problem with "logic" is that there is no one, true system. There are many possible logics and at a certain level of complexity they are more or less equivalent (a Turing machine, for example) It's a just a system of symbols that we refine to represent our world. After we spend enough work inventing a powerful symbolic system we can get confused, turn it upside-down, and start thinking that it has some inherent meaning. That's silly.
But there are systems, and more to come, to test against whatever these supposed to work with. That doesn't invalidate platonism, only tells that models may not match something. But that is already obvious, like you cannot win chess by poker rules — they do not even match the problem space. A spherical infinite set of eternal mathematicians could think of and explore all of logic systems. What is a true system anyway?
> Logic is fundamental and proven to be able to apply in reality
Nothing is fundamental in logic, 2500 years later and we still haven't "figure out" Zeno's paradoxes, and as such we haven't been able to refute either Parmenides's nor Heraclitus's way of looking at things/reality.
Yes, the statements brought up by Aristotle a little later allowed us to be pretty efficient further down the road (without Aristotle's "Logic" we probably wouldn't have had "science" as we know it), but there's nothing fundamental about them. And I suspect Aristotle himself knew that, almost the only way he could attack Heraclitus's views were "ad hominem"s, there was nothing "fundamental" in his explanations of why Heraclitus was wrong and why he was right.
Zeno’s paradox is solved in a better framework for that problem.
Maths is about creating an abstract framework with the mind (which somehow exists without typical bounds of time and space like everything else we can touch). Then you use the framework to sometimes solve real problems.
>Logic is fundamental and proven to be able to apply in reality.
Logic only exists in our minds as well as other things mentioned. The problem which arises with making logic fundamental is that it does not do much for things which matter most in our lives. Like logic can not bring you happiness and joy. I can't make your experience sweet or ecstatic. Nothing wrong with logic it's just limited. Once people see those limits clearly they start exploring life beyond logic.
I recently read an article on HN about how smart ravens and crows are. The connections in their brains are wired differently than primate brains. I wonder if their logic is different than ours for that reason.
The foundational crisis of mathematics in the early 20th century should have laid the notion of Platonism to rest, right? I guess a counter-argument to that could be that the Forms would be incognizable.
In either case it's hard for me to see what Platonism can tell us about the world or ourselves given all we know about math and it's many gaps.
One trouble with debates like is math invented or discovered is people will call 'discovered' Platonism and then get into debates on all the miscellaneous beliefs Plato may or may not have had.
This is wonderful. I love when memes breed philosophy. We're teaching math wrong. It's as plain as day to me. Or rather we teach the wrong maths to the wrong students. If Marcie can't find a useful application on day one of the "theory" she learns in school, what is the point? It's a 19th century curriculum for a 21st century world. Basic accounting and personal finance would probably serve most students better.
As to the epistemological quandary, I always return to Geometry or Visual Mathematics. Consider the infinite ways of calculating the digits of PI. In "Playing Pool with PI", Galperin demonstrated an analog computer that could calculate the digits of PI using only the perfectly elastic collisions of two billiard balls on a pool table. It works for all integral base number systems, and can even be extended to irrational numbers. And recently Google Brain's Adam Brown made an equally stunning observation: Galperin's bouncing billiard balls look identical to the Grover Quantum Search algorithm!
Playing Pool with |ψ⟩: from Bouncing Billiards to Quantum Search
Personal anecdote: I never understood what math "was," and this totally demotivated me to study it. I got A's all the way through high school mathematics because I was a systematic and logical thinker who had no problem following rules or algorithms, but I never once saw the point. When I reached college, I had already tested out of Calculus and never took a formal math course again.
Fast-forward ten years. I am a self-taught programmer and now find mathematics absolutely fascinating and beautiful, and I see its applications. So I'm taking math classes on Udemy, buying hundreds of dollars of textbooks, and trying to teach myself.
I can't help feeling like I got a little robbed. I understand that teaching is hard, and some kids just don't care to learn certain things and it's best for them and society to force some absolute fundamentals down their throats. But at a certain point around middle or high school, I think there needs to be a hard pivot to focusing on motivating students. If you do that well, they will have the ability to teach themselves and learn for the rest of their lives. If you don't they either end up like me, playing catch up a decade too late, or they just leave entire fields of knowledge untapped forever. With some subjects that's inevitable and probably ok, but math doesn't seem like it should be one of those subjects.
1. A clear explanation of a symbol system and how math is a symbol system to deal with issues of quantity, shape, structure, and logic.
2. Frequently converting between diagrams/drawings and symbols (show "x x x" at times and "3" at other times). I actually have a concept for a game I want to program that does this.
3. Showing multiple concrete applications of different ideas, and lots of word problems. In retrospect I think this was always reversed in the American system. We would learn the symbolic and algorithmic aspects of a math concept, then use it to solve some word problems. Once I went back and started teaching myself some of this stuff, I read that the Russian system is quite the opposite: lots of word problems, multi-faceted problems. I suspect I would have learned from that style a lot better.
EDIT: One more point I should make. I scored in the 85th percentile on my SAT math. It is probably not a great sign for our system of education that I could do so well with nothing but the most rudimentary understanding of what I was doing.
Interestingly, much of what you describe is present in old mathematical texts. For example, Fibonacci's Liber Abaci is essentially a long succession of story problems. Diophantus's Arithmetic is a succession of concrete problems -- no story -- but it's concrete prior to abstract as you suggest.
Euclid II also develops most of Algebra geometrically, so there's much conversion between shape and symbol baked it. His method of computing the product of binomials is far superior to what's commonly taught.
So there's a good tradition of the thing you're looking for!
I liked math in high school and usually saw the underlying meaning of the concepts. Teachers generally aren't idiots (or at least mine weren't), they try to explain it best they can, and start with that explanation. I remember explaining the meaning to friends that had only tried to memorize the rules without bothering to consider what they meant. They were surprised to learn these meaningful explanations, but in fact I was just repeating what the teacher had told us. I think the difference is I came into the class knowing what I should, while they had gaps in their prior teaching that they had to compensate for with memorization. So I was ready to hear the motivation at the start, but others weren't ready yet until they had memorized the concepts as a kind of crutch.
What you want is methodology like "just in time learning" or "problem based learning", which is very powerful but also very slow. You have to greatly reduce the content you can cover in a class. It's good for some students, boring for the smart ones (who solve your fancy word problems instantly), and the worst ones don't pay attention or do homework either way.
I'm not really convinced that's the case. As I mentioned, the Russian system is apparently very much like this, and they don't seem to have a problem producing large numbers of advanced quantitative thinkers. I also take a little exception to the implication that I wasn't one of the "smart ones." I had no problem doing what I was taught, I got good grades at at an advanced prep school, and I scored just fine on standardized tests. I basically succeeded according to every standard set by the system, and yet I wouldn't say I really learned very much. To me that signifies a real problem. I certainly deserve some of the blame, but definitely not all of it.
You're not convinced of what exactly? That alternative methods reduce content? That is precisely what you are instructed to do by pedagogy researchers giving advice to teachers.
Asian education systems like India, China, Japan are very rigid and based on rote practice and memorization. "Prep school" there means evening schools where they just go do yet more memorization and rote practice to help pass entrance exams. Yet they produce many brilliant mathematicians too. Of course the students are motivated by parents and society, not love of cramming.
The ideal would be teaching that is individualized and does the best it can to get through to each student (there are plenty of efforts and even "edTech" that tries to do this, especially at lower grades). Without that it's always a trade-off in which students get the most value for the time spent.
You said it's "boring for the smart ones," which I took to mean that if we go by the methodology I'm suggesting, we'll bore all the smart students, they'll have less knowledge, and we'll produce fewer good ones. Maybe you didn't mean all of that, but if you did, my point was that the Russian system, which seems to operate this way, seems to do just fine in keeping good students motivated and producing lots of talented mathematicians. I'm not convinced volume of content = engagement and better outcomes, either for "the smart ones" or the rest.
What convinced me was the book Goedel, Escher, Bach, by Douglas Hofstadter. It was my first exposure to the more abstract and playful side of math, and motivated me to become a math major in college.
I was "good at math" in high school, to the extent that I could crunch through any problem in the math or physics textbooks. School math existed mainly to serve the science and engineering students. Now I was happy to use math, for instance in my electronics and programming hobbies, but that wasn't math as an end unto itself.
But proofs are what really made math come alive for me. I don't think this makes me a freak. A lot of people I know from my generation said that their favorite high school math course was geometry, which was heavily proof oriented at the time.
Sadly, the contemporary K-12 math curriculum is sorely lacking in proofs.
Now, what about the students who would be more motivated by usefulness than by abstract theory? For those students, I suggest looking at how people in the so called "real world" actually do math, and work backwards from there. My days of abstract math are behind me, but I am still one of the "math people" at my workplace, in an industrial R&D department. When I do math, I'm never far from a computer. Yet school math is still taught by mainly pencil-and-paper methods, even if those have been translated to online forms.
For many of my math tasks, I start by playing with numbers, e.g., computing a function and graphing it in a Jupyter notebook. It brings tears to my eyes that K-12 students are not exposed to this. For one thing, it's fun, and it's an honest portrayal of how people actually work. For another, you can brute force your way through a problem even if you've forgotten some particular formula, making it more likely for someone to use it later on. Graphing calculators are of course a thing, but they hardly go far enough.
Interestingly (I'm one of ther guys self-teaching as an adult), this is exactly how I'm learning. I'm doing a course that uses Jupyter to cover all of the major areas up to linear algebra. I also aced my Geometry class in high school exactly because of what you said: it gave me some of the flavor of proofs and "real math," which was fun and made me motivated. I remember at the time being sorely disappointed when no other math classes were like that again and figured it was some kind of lucky fluke.
I think you're onto something, because teaching myself programming was precisely my backdoor into math. I remember a line in the Clause Shannon biography "A Mind at Play" that said he was essentially the same way: he wanted to apply his mathematics and understand the proofs/structure behind it, but essentially (as the title suggests) it was a form of play.
I'm glad you wrote this comment since I think this happens to almost everyone in the current system to some degree. It's sad that even those who see the beauty in mathematics are basically forced to rote learn algorithms to recite them later. It's no wonder that maths, school, and boredom are synonymous for a lot of people.
> If Marcie can't find a useful application on day one of the "theory" she learns in school, what is the point? It's a 19th century curriculum for a 21st century world. Basic accounting and personal finance would probably serve most students better.
Sorry, but following this logic we will end up with a conclusion that almost no school knowledge is "useful". How useful is playing volleyball? Reading poetry? Painting? Studying history? Biology would be reduced to first aid and basic medicine.
I have those discussions with my daughters who learn how to do long multiplication. How is it useful if even I who know how to do it would rather use a calculator on my mobile instead? We would reduce math to basic operations on a calculators, probably nothing more would be considered useful by the majority of the population. The same apply to all other subjects.
I think "application" is the wrong focus. The real thing to help students understand is this exact question: what is math? That remains a mystery to most students. It gets conflated with the problem of application, but I think the real problem is most educations don't make clear the idea of abstraction, symbol systems, and logic. When you don't understand something, especially why it might be useful or at least intrinsically interesting, it's hard to get motivated about it.
But this exact question has no answer or has many different, contradictory answers as this discussion shows. I doubt that such discussion would be interesting for someone who struggles with algebra. Moreover the same questions could be asked about all subjects. The point of the school is to show kids different things even if they cannot see their usefulness right away. One person will enjoy math, other poetry or history. But all will have a chance to try different things and to choose what to study deeper. Of course it's nice to give some motivation and her teachers failed if she cannot see applications of basic algebra.
I responded to the comment above with some examples of the kind of thing I'm hoping for. I don't think a deep philosophical discussion about math is necessary. Just reinforce the idea that math is a symbol system for dealing with questions of shape, quantity, and logic, and give multiple examples of applications of the same concept. Also convert between symbols and diagrams/drawings frequently (for example, I'm willing to bet an absurd number of Americans can't understand how a virus spreads exponentially; if you draw a pyramid of infected people, I bet they'd get it immediately; but instead we use curves and lines that don't connect with people).
What is interesting is I'm not sure how many commenters of the video listened to it. The actual points she raised were mostly focused on "why would Pythagoras have used algebra?". And the answer as far as I can see is he probably didn't. Most of the notation we use was introduced in the 1600s-1700s [0].
The concepts we are taught today, and the way we are taught them, would be foreign to Pythagoras. According to wiki, symbolic algebra only emerged in 1200+ with the Arabs (which probably links to why we don't use Greek by default for algebra but we do with geometry).
Pythagoras didn't use modern algebraic notation, but I don't think it's correct to conclude from this that what he was doing wasn't algebra. He reasoned about relationships between abstract quantities in ways that we today would write down in algebraic notation. Does a change of notation mean that the reasoning itself is no longer algebra?
The ancient Greeks did stuff by example, not using algebra. They said stuff like, "take 22 and multiply by 5..." with actual numbers involved, then at the end they say "and that's how it works in general". So it's up to the reader to generalize from the specific. Algebra works in the opposite direction, where the reader is obliged to specify from the generic. I happen to much prefer the ancient Greek way of doing mathematics, but it seems to have fallen out of fashion.
There are no concrete quantities in sight. I don't think that they were removed by the translator. Much more plausibly the original reasoning was symbolic in the way indicated here. Though again, possibly using different notation.
Ah, you might be thinking of https://en.wikipedia.org/wiki/Diophantine_equation. Sure enough, https://en.wikipedia.org/wiki/Diophantus says: "It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus. “Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions”." I didn't know this, thanks.
This is a rather interesting article. I'm a Platonist. From what we know today, real numbers cannot exist in a finite space, but they seem to exist mathematically. The same can be said about many other mathematical structures, including those that can only be characterized adequately (categorically) in higher-order logic.
It's also worth noting that physical and mathematical existence are based on completely different criteria. For non-constructionists mathematical objects exist once they are not demonstrably contradictory (although the absence of contradictions often cannot be proved in an absolute sense). In contrast to this, for physicists an object exists once it can be measured, where measurement is ultimately tied to sensual experience. There are also theoretical entities in physics that cannot be directly measured, but their existence is usually downplayed, they're not supposed to "really" exist but only as theory-dependent entities. In any case,the two "kinds of existence" are very different from each other.
The perception of numbers exists mathematically. Likewise for more abstract structures.
You cannot show me threeness. You can show three things and tell me to generalise, and if you keep generalising you'll end up with a consistent-ish framework of sorts for your observations that can be applied to certain other physical experiences.
But fundamentally this is an exploration of the consequences of psychological processes - like perceptual grouping, and inductive relationship inference - not observations of external phenomena.
It doesn't seem that way, but there is no external authority you can appeal to which will state definitively that when mathematicians all agree on something their experience of "true" is absolutely and objectively correct, and not a distorted and limited artefact of human cognition.
This kind of sensualism was often defended in the debate, but it has problems, too. I think the history of mathematics makes the position implausible.
For a long time, up until recently, mathematics was way ahead of the applications of mathematics to physics. In Ancient Greece there was a general consensus that infinity and real numbers do not exist. But then some people found out that the side of a triangle must sometimes be a real number. However, you cannot ever measure SQRT(2) precisely. Whatever number you extract from the physical world is only a rough finite approximation. You need to represent the number in a different way and solve the problem algebraically. Many scientists in Ancient Greece rejected this idea vigorously. Still, real numbers are very useful for describing the physical world, so useful that we couldn't possibly do without them today.
Imaginary numbers are another example. They were ridiculed as abstract nonsense when they were described for the first time and widely conceived to have no physical reality or application at all. Despite all that, they play a vital role in modern physics.
There are many more examples like that. To cut a long story short, at least until recently mathematics was always ahead of physics (now they seem to go more in tandem). This fact makes the idea very implausible that mathematical structures are merely useful abstractions from the physical world we invented to describe it. It simply doesn't describe what happened in mathematics. And I find the idea that mathematicians just came up with arbitrary imaginations equally implausible.
> no external authority you can appeal to which will state definitively that when mathematicians all agree on something their experience of "true" is absolutely and objectively correct, and not a distorted and limited artefact of human cognition
That is true for everything, it's just a radical skeptic position. Nevertheless, mathematics has the highest standards of rigor for proofs among all disciplines.
Quite an astute observation. I’ve been thinking about this quite heavily recently. It has odd occult references too in the sense of “nothing creates something.”
That which is measured, exists.
No measurement itself, as in of itself as a thing, is a thing that exists. You can measure 3cm but “3” and “cm” don’t exist. Virtual values assigned by consensus. All axioms are agreements. Truth or existence itself is convergent and resembles certainty only in the majority agreeing to it.
Imagination allows nothingness (imaginary things) to be measured.
Imagine any unit, give it a “rule” and now it can be measured!
Pretty interesting bridge between the “imaginary” and the “real.”
There’s so much nuance on what existence is, and isn’t, or even not is nor isn’t is depending on if you subscribe to classical logic based on what we currently describe as mostly Aristolian or if you use many-valued logics which are making a come back yet rooted in the Vedas...
I find this extremely profound and it’s one of my top focal areas of study right now.
It’s also related to my observation of disagreement, chaos, and behavior of the human animal. It seems the zeitgeist of the egregore lacks the ability to adjust perspective and see truth depending on the “rule/roles being played” (all disciplines are games/acting in a way).
In fewer words than the article, there's a concept called a substrate, kind of like what water in an oocean is to a fish, and while we experience the patterns of it all the time as part of our normal existence, expressing it in language that lets us share our experiences of it with each other is very fiddly. But when we manage to describe things that consistently reflect other peoples experiences, it is like drawing a map. You can experience the territory without it, but it's a totally different game when you have one. So between the critical theory poles of empiricism and Platonism is our ability to perceive and express our experience of our substrate.
(What is our substrate? Turtles, all the way down.)
Institutions these days never seem to pass up a teachable moment when it comes to reinforcing the precepts of dialectical materialism. e.g. "If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)" It's so tedious as to make me doubt their humanity, but I'm glad this girl sparked a popular discussion.
I particularly like the closing comments about how we teach math. It is something of a theme that we do a bad job of explaining why math should be taught. It took me many years in academic research to realise that not only was math useful to me, but that I liked it. If we could do a better job communicating what math can do - make us rich, predict the future, win at games - maybe it would hold the attention of our students better.
One could argue that a defining characteristic of human intelligence is our ability to agree on "unreal" concepts. I think the first time I heard it explicitly was in the book "Sapiens", though I doubt it originated there. In this sense, mathematical concepts are no different than, say Santa Claus. We can agree (mostly) about what it is we're talking about, and talk about it. What sets mathematical "unreal" ideas apart is just how useful they seem to be in describing the universe. They're patterns that fit onto lots and lots of things. For instance, we can all sort of agree on the idea of a (finite) set, and on some basic rules of logic, and with those you can build a lot of mathematics.
I was disappointed that the article didn't describe some of the motivating forces behind the basics. I think such concreteness would make more sense for a high school student. As an example, the development of geometry was (at least partly) motivated by the need to restore property lines after the annual flooding of the Nile in Egypt. I have no idea if it is part of an "origin" story for probability but wanting to win at dice games is one motivation for understanding probability. One of my heroes, Grant Sanderson, the creator of 3Blue1Brown is very clear on how the concrete needs to precede the abstract in many of these studies, and y = mx + b is pretty abstract if all you've got is those six symbols.
Of course, the whole platonist/realist/phenomenologist (sometimes characterized as the 'discovered' vs 'invented' argument) is hardly new, nor likely to be resolved (if that were even possible). I certainly recall an enormous amount of cheap on-campus beer being spent exploring the issue when I was an undergrad 4 decades ago.
I did find a couple of things quite refreshing in the article, though: (1) Wigner's essay is delightful and I'm glad it's seeing a wider audience. We should be reminded of it every time this conversation comes up, if only for its last paragraph. (2) The reference to the fact that Galileo proved that falling objects accelerate at the same rate, not by some experimental demonstration, but wholly within his mind---and it's a proof that any middle-schooler has no difficulty completely understanding. As a reminder that mathematics isn't symbols and rules, but a way of looking at problems (Eugenia Cheng's book, How to Bake Pi, is an excellent look at this way of thinking and approaching the world.). (3) It managed not to get bogged down in the whole pointless and hypocritical "what is math good for" instrumentality debate, that these sorts of questions tend to devolve into.
One could equally ask what is history, art, sports, etc. The answers mathematicians give seem partly interesting, but also, to me, imply insecurity -- that they have to justify to others that everyone should learn it. Not many other subjects suggest needing justification. Meanwhile, kids teach themselves math in self-directed learning environments https://www.psychologytoday.com/us/blog/freedom-learn/201004....
Up to high school they taught it to me by textbook, mainly teachers who didn't know or care about it. I found beauty in it and loved it despite them. I suspect if we taught it less relying on authority and grades, more engaging and based on curiosity, we'd appreciate it more. Instead of asking what it is, we'd do it.
Even people who don't play music don't ask what it is, like it doesn't need to exist. They probably would if we just taught music theory from textbooks but not singing and dancing.
Kids just have no motivation for math. It's not obviously enjoyable like art and music. Teaching what it is and how it helps them in the real world motivates them to learn.
It pains me to repeatedly see debates which were decidedly answered in the 1700s by Kant in the Critique of Pure Reason--with such persuasion to even turn Hume. Transcendental idealism is more or less non-controversial in mainstream philosophy. Especially without even a mention of idealism from the author of the article.
Math is a synthetic a priori judgment, a transcendental object.
I can only believe that this fashion of thought is from the unwarranted development of the nihilist project which came nearly 100 years later.
Bertrand Russell wrote in his historical treatment of philosophy that much of philosophy is forgotten and rediscovered throughout history. I'm skeptical of the argument but less so everyday.
>Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist...
Roger Penrose states in his book "The Road To Reality" that he is a light Platonist.
It pains me to see people who are blithely unaware of the philosophical challenges that mathematics faced from repeatedly falling apart in the 1800s and therefore believe that the real challenges were already solved before that happened.
It's a straw man to argue that math has brought new philosophical challenges, no one has posited against that. I've only stated that math is readily understood by transcendental idealism, disproving the false dilemma presented by the author of the article.
A more mature comment would be to point out an instance where there is some philosophical challenge to transcendental idealism that came from 'new mathematics'. Though I find the possibility of this to be impossible. I would expect at least the German-Idealist Godel to write about it.
If you're talking about Gauss's non-Euclidean geometry, that's completely compatible with transcendental idealism, in fact Gauss took his non-euclidean geometry from the second chapter of The Critique of Pure Reason. He read Kant's passage on the triangle and the line in respect to the sum of angles obsessively.
I'm talking the collapse of infinitesmals as a basis for Calculus, the development of a more rigorous foundation, the rise of axiomatization, debates around the validity of pure existence proofs, the failure of naive set theory, and so on.
On pure existence proofs, mathematicians have generally agreed that something can be proven to exist by proof by contradiction on an infinite set. The result is that we say that it exists even though there is no way to find it, and no way to verify that you have found it if you are presented with it. Nothing in Kant's philosophy had anything useful to say about the debate about whether to accept this kind of proof.
Thanks for your reply. The first part, although mostly true (axiomatization isn't new and Kant was well aware of Euclid), is wholly irrelevant so I'll comment on the second part.
None of this contradicts the synthetic apriori of mathematics, in fact Kant's critique strictly tells you that analysis IS possible and guarantees its legitimacy. I would look to Wittgenstein and Godel for that expounded application of the transcendental idealism.
If you state that philosophers mostly have a consensus on anything, you'd be brutally wrong. That's not how philosophy is, or can be. Consensuses is not what they do.
Indeed this is precisely the unwarranted nihilism I'm criticizing, where if we were to believe that consensus in philosophy is impossible, so would philosophical progress--and therefore knowledge. This is a libelous claim that all philosophers are engaging in intellectual masturbation explicitly aware that their exercises are to be of no progress.
The Pyrrhonists, Diogenes, Descartes and others famously believed in this; but Kant fixes this, making the discipline of philosophy possible by proving the scope of reason.
We have taxonomies of objects and concepts, but do we have taxonomy of relationships?
This might continue to mystify us as long as we try to push the square block into the round hole. It worked so well with the smaller pieces..
Every new discovery started with a question and letting go of some assertions.
The article spends most of its words discussing the reality of math via Platonism vs empiricism while forgetting its own title "What is Math?"
If going for a pithy answer, I like tha following quote by G. H. Hardy:
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
I think I originally saw this quote in "A Mathematician's Lament" [0] (also known by the author's name as "Lockhart's Lament"), which has a lot to say about modern math education.
I like to think about it as this: numbers are a human invention, but quantity is a property of the universe. The laws of physics, including all types of conservation, take care of that. Quantity takes care of ensuring many derived properties (rules) that numbers show and which can be expressed abstractly (complex math). Studying the properties of a complex self-consistent system is interesting, and sometimes, when it reveals an interesting resulting property, it can be traced all the way back to the original natural property of quantity, and applied in a practical way in the world. Other times, it can't, and it remains a curious interesting fact in the realm of the abstract system.
I did a thought experiment few years back: Is anything real if we live in a simulation? I came to conclusion that all physical laws, chemistry, the universe would be simulated. And the only "truth" that is the same for host and simulation is math.
Imagine that you're creating computer game - a limited simulation. You can change laws of physics, chemistry, life, etc... but math will always hold true. 1 + 1 will always be 2.
It’s important to note that no model using math has accurately represented nature. Just because simple approximations line up in a beautiful way is not reason to think they are 1:1 the same. Math is not sacred or divine. It’s a tool. Like all tools: its shape matters.
Math, as we know it, is invented. It is one set of rules that we like to use to describe things. It is not the only way. I can imagine an intelligent species that has a structured set of logical rules that looks very different from our math but still be able to make useful models.
The symbols may be invented and the things we have investigated may be different, but the things we prove are true, even to the aliens.
Consider modeling the game of tic tac toe. You can prove that certain strategies always win. If you then proceed to play it with that strategy against the aliens, then you will certainly win the game, regardless of how their logic looks. (this assumes you're starting of course)
That just means math is a set of self consistent rules that you could teach to an alien, not that they are the only set of rules that are self consistent or the only rules that can be used to model empirical observations of nature.
Said another way, if you have a dictionary translating, it doesn’t mean the definition is equivalent. In fact, most definitions aren’t, as each one has its nuance. If following a dictionary to arrive at a result after some steps, it doesn’t imply anything other than a set of rules being followed to arrive at an arbitrary value.
Universe as one mind depending on zoomed out vs zoomed in.
However this attribution of mind is simply anything that follows rules. So it’s better to call it “logics” and humans have a property of “bio-logic” abilities. But then what exists outside of logic that allows arbitrary definition of things to represent said logic? The programmer, the consciousness. In general I feel mind exists outside of logic but that’s perhaps a matter of perspective and definition. It gets hard to know where one has defined a conceptual space between continuum and borders and assigning it a range of values.
Rules don't exist without consciousness assigning a value. Which is to say that consciousness arbitrarily defines truth from nothing.
Rules are not what they are because a rule doesn't have any intrinsic property outside of imagination. It's dependent entirely on consensus and memory.
Without communication, consciousness wouldn't be able to distribute rules. It'd be limited to one mind per set of rules. Communication allows imaginary network effects. Distributed protocol message passing, yay!
Also to add, just because the rules are followed doesn’t mean the proof was communicated. Any layer of translation can act in between two entities without understanding.
That's very true but I also think of one important aspect: the parts of the research space we choose to look in. And like a novel is not discovered in the absurdly large research space of sequences of random characters, nor even discovered in the absurdly large set of sequences of grammatically correct sentences, theorems are made were "we", concrete social creatures, look for them. But there is a distinction in that one activity lies in a formal logic world, the other is more free (neither formal and can even have small inconsistencies)
So maybe we just need yet another word? Discovented?
The universe exists as a geometry. That geometry has properties with different interactions. The geometry exists regardless, but we can give that geometry a name arbitrarily.
It’s also the difference between music and math which is equivalent to me as frequency and quantifying frequency.
Vibrations interact with each other. Mathematics is the observation and labeling of interaction(s).
Tricky because vibrations are continuous and now we’ve got a can of worms on continuous and non continuous numbers. Haha.
“A 3×3 game is a draw. More generally, the first player can draw or win on any board (of any dimension) whose side length is odd, by playing first in the central cell and then mirroring the opponent's moves.”
I don't think it does, but I am not sure. Given how easy it is to draw normal tic tac toe, it definitely seems more interesting to flip the win condition.
but I believe anything is true if embedded in a metamodel making it so. I mean aliens might think in hyperwave cofields from the get go and to them our atoms might looks totally absurd.
I would argue that math is both invented and discovered at the same time. Our expression math is obviously invented but that's not very important (but not totally unimportant either, because it drivers further discoveries). The set of axioms we chose is way more invented than discovered, although we won't like others, it is likewise we won't like e.g. a too random or too complicated a mechanical apparatus, or even it would be useless. Given a chosen set of axioms, the theorems we can infer are so constrained that the only reason we can find an invented component is that we chose the places to look for in an absurdly large research space. So like we can argue that we don't "discover" a novel in the large research space of random characters, we can in a way argue that we in part invent the theorems.
In the end does it really matter though? Is it maybe not just that inventions and discoveries have more things in common than some intuitively think? Tons of people are not really found of patents, or at least of some class of patents, because the line is arguably blurry (so yeah there is a practical way in which case it do matter, but only because of a completely arbitrary social organisation)
That isn't an approximation. It is a core principle, not just of math, but of the laws of our universe. It is so core, that it's impossible to imagine a universe without it.
or maybe it is an approximation. Maybe there is some essence of the universe that gives is additiveness - and maybe if you could see closer/bigger/faster, maybe that too would break down.
Now you're cheating, because + and - no longer mean "addition" and "subtraction", but merging and... well, weird division - the last example would be better as 1 stick / 2 = 2 sticks.
But try the first two with the following substitutions:
You added additional axioms that require a quantity. Not all things exist as quantities with properties of addition, multiplication, division, or subtraction. Various debates can be had that any quantity is largely an arbitrarily defined border.
The point mostly being, any symbolic representation is only that. It is never universal. And to think it’s an objective representation of anything is serious error.
Sure we can measure something else to keep a track of whats happening and fit our definitions, but the math remain true, isn't it? It is still 1 river and 1 drop and 2 sticks.
You are correct when you say its not a defined measurement. And that is the point. Not defined in math.
Math is our reality, and not a universal reality/law/principal (while science conservation of energy/matter is universal). There are no addition/subtraction happening in the universe.
Addition is not conservation. But that's how we see it.
It’s one of something. When one of something and one of another thing are observed it’s expressed as 1 and 1 so two rivers become one they represent a quantity of observation.
Then as 1 and 1 totaling two rivers they come together then exist as 1.
Representation is relative on an arbitrary border of quantity. The rules leak for any rule that exists.
One father and one mother have a kid. One and one make another one totaling three.
• and • make •.
Usually a person will substitute 1 for the observation of a “thing” so • will be father, and another • mother +1+1=+1
One unit and another one unit makes a third unit. Yet one unit plus another unit is two units.
> It’s one of something. When one of something and one of another thing are observed it’s expressed as 1 and 1 so two rivers become one they represent a quantity of observation.
Not really. If I look at a map and see the Amazon river, and then see the Nile, Amazon + Nile ≠ 1 river. They're still two separate rivers. Adding rivers means merging sets of rivers under consideration, not merging actual rivers.
All you're doing now is playing with different operations that are not what the symbol "+" stands for.
I don’t think that you’d be able to communicate the observation of units in a way that’s consistent as a written form of symbology in the rule space of mathematics and the language space used to communicate the mathematics and observation of reality around oneself which leads to relativity.
All definitions are nuanced and subjective based on path of least resistance heuristics.
It’s often miscommunication is happening because expected results are misaligned despite using same rules/language.
It’s quite complex. It’s really interesting but also leads to poor communication. Do we try to embed in our cultures to align on expectation and values to improve this or always doomed to potentially fatal inconsistencies? I guess the dynamics of any rules naturally incentivize behavior whether it’s negative or positive.
1 does not equal "1 of <something>", the latter is a concrete example only (and as mentioned the quantity error, you would have 2 drops of water if the volume of 1 drop stays consistent).
1 is an abstract concept applied to many things that share this "singleton property".
I disagree, computation has equality, unless you think computer files are not real. Two computations can be equal, and as far as we know we are leveraging this property of the universe. Electrons move and produce patterns which have equality and gives rise to computer state and thus equality of values, not to mention electrons themselves must be indistinguishable given they behave the same.
Yeah except it’s not that easy. 1+1=2 but could 1+1=3 too? Yes, even
There’s different ways to count based on the dynamics of the system you’re interacting with.
One interaction interacting with another interaction can create a third interaction causing the system to interact as as pieces totaling 3 that came from a 1 and a 1.
1+1=3.
Also, my brain may be split into two spheres however is it better to reason about it as 1+1=2 me’s or 1^2? Funny joke if I use i^2 then I’m just imaginary. Hah.
Ultimately anything we describe as “is” isn’t anything other than agreement of some type of axiom which requires consciousness / imagination.
In that raw written symbolic form, perhaps. However the nuances exist where the symbols don’t apply but the language is still used. You can observe effects/interactions and say one thing added with another thing makes three things. There is not two things, there’s three things happening now. Also one can be added to another and make one. The “adult” costume for two kids hiding in adult clothing. Two became one interacting as three as a system represented by one. The values are relative. There’s a danger in having such zoomed in views. On one hand we must know when to zoom in, and on the other know how to capture vagueness.
That isn't really about mathematics anymore. With the meanings of the words/symbols that koheripbal intended, 1+1=2 is definitely true, and similarly, with the meaning koheripbal intended, 1+1=3 is definitely false.
If we can't agree on what the words mean, we can't really have a conversation.
That’s kind of the kicker. Many are communicating all day every day with the assumption of equivalence to mappings of words to abstractions and largely get by but it is inconsistent.
More or less consistent enough is certainty.
Nuances exist, as well as unknown side effects.
When has any definition you thought were fixed expanded?
I like this question. Addition is closer to experience sounds interacting. For example one person playing a note then two notes at once or maybe two instruments start interacting. Either way, it’s our experience of some dynamics based on observation.
We really don’t know that to be certain (the fact that math is invented). This is a very longstanding philosophical debate with arguments on both sides.
Now, of course the symbols of math are invented. But math is the study of mathematical _objects_, and it’s very unclear if those are invented or discovered.
For example, the Curry-Howard isomorphism shows that math proofs and computer programs are the same mathematical object. Does this mean that they are a part of nature, and we discovered them because we invented a sufficiently powerful mathematical system? Or are they simply properties of the invented system?
We do not (and most likely can not) know the answer to that question. If someone is extremely convicted when offering an answer to it, consider me completely suspicious.
This is something that hit me when I found out how to translate between decimal and binary. If an A on a screen can be a grid of numbers, and those numbers can be reduced to a series of 1s and 0s and stored in a bunch of fancy wires, then who knows what else it could be translated to or from. Look back far enough and the A is a series of photons clashing with orbiting subatomic particles at certain frequencies.
We use math based on the status of a sea of transistors to store it in other atoms with forces we're slowly discovering how to communicate about. It's all layers of finding new ways to describe something that was already there, spinning and bumping photons long before any of us had photon detectors in our heads or a chemical machine capable of deciding to call one pattern A and all the ways to translate it into sound waves.
One pattern is called "solid state drive," and we seem to have some say in what that pattern does from our frame of reference. Maybe it was always a solid state drive and we're the language the universe uses to describe its smaller parts.
I promise I'm not high, and probably not a Boltzmann brain. Though I'm not sure about the latter.
Signal theory is deeply interesting based on your line of thinking here. I’ve been postulating similar things. The universe gives a lot of potential for things to communicate. In the end what is recognized as signal and or information is a matter of imagination and consistency given the plethora of potentially unlimited signals available.
The education system has utterly failed in its duty in teaching how to use the mind. Instead we see masses are indoctrinated and told what is, and what to remember rather than first principles and tooling to reason.
Any particular point I should be refreshed on? I mean the uncalculated digits of pi obviously exist in the sense that with a better computer you could discover what they are regardless of what various philosophers may or may not have said.
But if it is invented then could we not have avoided inventing some of the more gnarly bits? If math is invented then, say, irrational numbers were a terrible mistake.
The building blocks of math are entirely abstract, but once we define a circle in any half-satisfying way then all of the results in algebra, calculus and geometry are largely set in stone. After the invention of the circle in pre-written-history, most of the math that followed was discovery of the implications.
I mean, irrational numbers are maybe "ugly" from a certain perspective, but they are the price we pay for other things that make our life easier.
Rational numbers are arguably nice and easy. But if all we have are rational numbers, 2 doesn't have a square root. More in general, our number line has holes, and one of the most fundamental properties of analysis - that every non-empty bounded set has a least upper bound - goes away. We could do away with that (and arguably, irrational numbers don't really exist "in the outside world") but it would make all the calculations so much more complicated.
The fun part is that what we invented is a set of rules that governs what else we may or may not invent! That is, once you agree on what logic is and what some axioms are, you can show that certain statements can't be added as axioms. Similarly, once you've decide on your axioms, and your set of logical tools, you will find some things (like irrational numbers), that you wouldn't have introduced directly. It's pretty cool!
No model can ever represent nature. A model helps prediction which helps convergence to certainty through consensus.
Although this opens a can of worms of what “accurately represented” means. A threshold comes into account. Then it unpacks into what is existence.
Fun stuff...
I’ve been thinking a lot of “errors” in our models (language, ideas, etc) are emergent from not considering change. Nothing in the universe is still. Everything changes or vibrates. The only things that are still, don’t exist.
It makes me imagine that we need an evolution in our language and core reasoning that intrinsically embeds change and or perspective (relativity?)
I’ve had this thought recently again based on studying what numbers are. The definition of numerals vs numbers vs integers and number theory, etc...
Even the “numbers” that we try to reference seems to exist as non fixed value of infinite change. Also makes me think of what is that core “thing” that allows a number to expose itself within our frame of reality? It seems to consistently show itself through mathematics and information theory and computer science. Perhaps all related to geometry in some way?
> I can imagine an intelligent species that has a structured set of logical rules that looks very different from our math but still be able to make useful models.
"Please don't comment on whether someone read an article. "Did you even read the article? It mentions that" can be shortened to "The article mentions that.""
Then they wouldn't be Platonists, would they? Wait, do people here even understand what Platonism means? Rejection of evidence is a key tenet of that philosophy. This is why the other school of thought is called Empiricists - because they believe in empirical evidence. It's frustrating getting all these downvotes by people that don't understand their own words, but I suppose it's not surprising coming from Platonists. Folks who, when faced with a contradiction between reality and their cherished beliefs, conclude reality must be wrong.
Neither the Britannica article on Platonism https://www.britannica.com/topic/Platonism or the Wikipedia one seem to mention rejection of evidence. I think maybe you have an unusual take on Platonism?
The important point is that physics will likely never find a perfect 1:1 correspondence between nature and math because of the properties of nature (such as being too complex for the human mind to fully model), not because of the properties of math (such as the incompleteness theorem).
That is to say, even if the incompleteness theorem hadn't been true, math still wouldn't have described nature with perfect accuracy.
Not to mention, the incompleteness theorem hasn't prevented any kind of modelling in physics that I have ever heard of. Indeed, physics is often not entirely constrained by formal methods, with ad-hoc mathematical models that can be shown to work even if they are not fully formalized sometimes being preferred (such as the Dirac delta "function").
Poets, like writers, use words. They often say things in few words subject to many internal constraints, so one must pay closer attention than when reading prose. (Indeed, many people write prose about the poetry as a guide.)
Mathematicians use even fewer symbols to say things subject to even stronger internal constraints, so one must pay even closer attention than when reading poetry. (Indeed, almost all people write prose around the symbols as a guide.)
Some poets explore what happens with more, some with fewer, constraints.
Some mathematicians explore what happens with more, some with fewer, constraints.
(Viewed as a MMO game, mathematics introduced the parallel postulate in or before 300 BC and although many people said it ought to be nerfed, the nerfing wouldn't happen until 1830.)
Platonism isn't the only way to belive that mathematical truth exists beyond humans. In Platonism one believes that the objects themselves are real in some sense, but I prefer to see it in a different light: mathematics is about considering the relationships between arbitrary objects. The actual objects don't matter - what matter is how they are connected, and the things we prove are true whenever a collection of objects fit together with that relationship.
> mathematics is about considering the relationships between arbitrary objects. The actual objects don't matter - what matter is how they are connected
You know this is Category Theory I assume? I guess it does describe math abstractly, but in a way it is also math. Not sure tbh.
This debate is partly a semantics game of the meanings of "discovered" and "invented"
If an idea describes reality does that make it a discovery instead of an invention? What if you think the idea describes reality but it really doesn't.
For instance, let's use the sense of inventing as pulling something out of thin air, and discovery as growing an understanding of what is in front of you.
Math certainly involves elaboration according to fixed rules which could be seen as discovery. It also involves what-if anxioms that redefine the basis for the mathematical thought (say 4-dimensional space), that could be thought of as invention.
This article seems to think of invented and discovered in terms of whether it is not based on reality or whether it is based on reality. This is trickier, I think it is fair to say that the needs of reality do drive certain branches of mathematics, but certain branches are proudly not based on reality. The funny thing is that periodically we realize that the math we thought was firmly not based on reality ends up describing reality better than the math we based on reality.
So you have math that was spun out of an idle what-if thought that ends up describing with high accuracy an important physical concept. Is that an invention or a discovery?
>Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist ...
That statement surprised me!
Google told me that according to this biography by J J O'Connor and E F Robertson [0], Roger Penrose in a debate with Stephen Hawking in 1994 self-described as a realist rather than a Platonist. They quote Penrose as follows:
>At the beginning of this debate Stephen said that he thinks that he is a positivist, whereas I am a Platonist. I am happy with him being a positivist, but I think that the crucial point here is, rather, that I am a realist. Also, if one compares this debate with the famous debate of Bohr and Einstein, some seventy years ago, I should think that Stephen plays the role of Bohr, whereas I play Einstein's role! For Einstein argued that there should exist something like a real world, not necessarily represented by a wave function, whereas Bohr stressed that the wave function doesn't describe a "real" microworld but only "knowledge" that is useful for making predictions.
Gödel's Incompleteness Theorem essentially thwarted Principia Mathematica by proving that mathematics itself is either consistent or complete (read: either useful or definable), but not both.
The best definition I've heard calls math "the science of pattern". However, proving itself undefinable seems fundamentally better.
IMHO Gödel's Incompleteness is also evidence that what is colloquially referred to as math is just a part of something much larger that we cannot yet grasp.
She asked two important questions, only one of which (the epistemology of mathematics)was addressed in the article.
The other one was “why were the Greeks (and contemporaneous Indians) wondering about geometry of a plane and why were the Arabs thinking of algebra” when they had more limited technology than we do?
my favourite (less than insightful) answer is that we live in a (hard) magic world and math then is the language of magic in a literary sense, which can then be used to formulate spells in like physic, chemistry, or psychology (we too are magic and can interact with the magic source in our own special way)
Platonists don't need to renounce empiricism. Assuming that I am real, if I can think about what mathematics is and arrive at answers, then logic exists and reality is non-uniform, it has some form of structure that permits non-uniformity of answers, allows questions, logic.
So the first thought I formulate is an empirical proof that reality has structure. Mathematics is nothing but a description of that structure of reality. We may describe it in human-centric terms and models, but the structure must by necessity pre-exist the mind and is empirically discovered by the action rational tought, not created by the mind.
No similar conclusion can be drawn for other non-empirical hypothesis, such as the existence of God. My ability to think does not imply the existence of anything else beside myself, so other empirical proofs must be considered.
What’s an empirical proof? I’m not being facetious, I’m genuinely interested because I’ve never heard the term before.
To me, a proof is a deductive mathematical argument that takes a claim of the form “if P then Q” and shows how Q is necessarily true when we assume P. I don’t know how to translate from reality into that formalism. All of modern mathematics does not attempt to do so. Instead, it is built up from the axioms of set theory, which take the form of rules for manipulating symbols on paper (or on a computer or even in our minds) without saying anything specific about what those symbols really mean.
That still leaves the definition of "real" and "exists". And even if a precise definition is given, that leaves the question of whether you are real or not. The answers to these questions are not incidental and are indeed relevant to your subsequent assertions.
> but the structure must by necessity pre-exist the mind
Why must the structure by necessity pre-exist the mind?
In other words, you are asserting that idealism (all reality is mind-dependent) is necessarily false, but what is your argument to justify that assertion?
(Even someone who believes that idealism is false might not agree that it is necessarily false, although the term necessity is ambiguous and can have different meanings – broad vs narrow logical necessity, metaphysical necessity, epistemic necessity, nomic necessity, real necessity, etc.)
I like that. The way I like to think of it is that mathematics is the language of the Universe. All parts of the universe can be described by this language but not all things described by the language may occur within the universe.
While it can be used to describe the reality we experience about our universe, it can also create imagined universes that have only the most tenuous grasp (if any). Like any language it supports fiction and non-fiction applications.
There are also rules about how the language evolves that are more strict and defined than human language and the evolution often is more akin to discovery. It’s like the exercise to try to write every single possible English language novel to find Shakespeare. Except instead of stories about human experience we’re trying to identify which random storyline we’re following maps to some fundamental truth about how the universe works.
You can sometimes also map fiction to our experience of the world so that it becomes more tangible (eg playing with 4d objects in a 3d video game viewed through a 2d plane).
This is no different from reading a fiction book to learn about concepts like compassion even though the story itself is made up.
So it’s not either or to me. Both empiricism and Platonism interpretations coexist. Empiricism is what we use to understand what story reality is telling us and how to confirm our comprehension of that story is correct. Platonism is that mathematics itself as a concept is something humans can explore and try to master but exists outside of us and the rules of the language are predefined outside of humanity. Note that just like any language it even has really significant gaps you can’t do anything about (which ironically the language gave us the ability to prove that about itself).
That being said the original question this article references indicates a lack of curiosity, which is where my problem with it stems from. And it’s not even like it’s hard to find (although it may require more than a superficial query). “What are practical applications of algebra” is answered in the first few hits. “Who discovered/invented algebra” can be similarly answered. These questions only sound deep if you are actively avoid actually trying to find answers to these questions. There are legitimate philosophical discussions to be had but the original question was not an attempt to have that discussion.
> While it can be used to describe the reality we experience about our universe, it can also create imagined universes that have only the most tenuous grasp (if any). Like any language it supports fiction and non-fiction applications.
Maybe I am wading into well-charted waters and thinking of something others have already hashed out, but if our universe only applies a subset of math, could that also mean there is another universe where a different subset of math is applicable? And the multiverse in sum encompasses all math?
Platonism would then be true at the most global scope, but empiricism is necessary to discover what part of math our particular universe encompasses.
How are you defining the word “universe”? Is it in a hypothetical imagined virtual universe? If so then yes, that happens regularly in mathematics (eg Carl Sagan’s flatworld). If you’re asking “are there multiple real universes in reality with some kind of relationship between them” we don’t know yet and until we have a falsifiable theory that withstands experiments we won’t know. Mathematics can be used to prove certain properties in the absence of those experiments and to explain how a theory might be falsifiable. An example of this would be string theory. Mathematically self-consistent, all available evidence seems to be against it (can’t recall if it was finally disproven or still just highly unlikely to be true)
Math is the series of statements that can be made about things that are true.
Math can't tell you that there are three of [anything that you encounter with your sensory organs], but it can tell you that if there are three of those, you can also say...
In that way, it tells you about the real world while having no connection to any real world. The problem is that as soon as a math-predicted observation fails, we can blame the mistake on our sensory organs - either formatively (when we assigned a particular observation to a particular mathematical entity (like "three"), or conclusively (when we assigned a mathematical result to a sensation.) If a mathematical result consistently fails to apply to any sensation, it ceases to be math - not by disappearing, but by failing to be consistent; if you can't apply a mathematical process to a sensation in a way that enables you to predict a subsequent sensation, there's no way to determine whether the math makes any consistent statement.
Math has to be empirical because it's completely vetted by sensational observations. If a hypothetical god changed some aspect of the underlying physical reality (inclusive of changing some aspect of human sensory organs), the math would change along with it, and things that were math would cease to be math. If you included that god within your mathematics, math becomes the set of all possible maths, which is just the universal set. There's no meaning to a math that says everything about everything.
I guess you can say Platonism is true too, since math is a statement about an underlying reality (because it is discarded when it is not), but it's not a meaningful thing to say. Math that turned out to be correct (and continues to turn out to be correct) is Platonic. Math that was incorrect (or in the future becomes incorrect) is not Platonic. The only way you can determine which math is Platonic and which math is not is through empirical observation, and the degree to which you can be sure that particular math continues to be Platonic is through how well your past observations predict your future observations.
To answer the tweet, people study math because because being able to predict things gives you an advantage in your work.
I like the idea of math being a fiction, but the difference between math and novel is that in math you can never ever break the rules once they're set. In contrast, I see that novelists constantly ignore or retrofit the rules to make it interesting. Imagine this was allowed in math!
I also think that the fascinating part of programming is that it's sitting in between math and fiction, depending on an application. Sometimes you have to obey the rules very strictly (e.g. writing a parser), but in other times you can be wildly imaginative (e.g. designing UI).
Well, people certainly invent novel proof techniques which remain controversial for over a century. For example, Cantor's diagonal argument, which people still deny the validity of today (although those who do are generally considered cranks).
The idea of something existing “outside of space and time” makes empiricists nervous
This seems like a boring objection: where does the letter 'A', or the ascii 0x41, exist?
I'm willing to admit that concrete instances of both exist in space and time (and I might even admit maps forming a galois connection between those and the abstract ideal 'A') but there's no "reference" canonical A, located in time, located in space, giving them meaning.
(Commented on someone but they deleted. Will post as parent? All axioms are imaginary. Given that humanities method of logic is imaginary, then consensus is the only difference between certainty and non certainty which is equivalent to truth.
Logic is programming. Human’s are programmers and hardware/OS who largely don’t realize they’re running arbitrary truth values based on popular imaginary rules.
> If mathematics is just something we dream up from within our own heads, why should it “fit” so well with what we observe in nature?
I like to think of mathematics as the formalization of human intuition. Insofar as that intuition has been shaped by our observations of the physical world, we should not be surprised that many of our mathematical models seem to describe aspects of natural phenomena.
I read that and was a little disappointed. He argues it's basically just a human invention and skirts around the obvious counter arguments that say aliens would discover the same maths if with different language to describe it.
He also tries to politicise the debate which I feel is kind of silly.
Also if you enjoy thinking about this topic I recommend Neal Stephenson's book Anathem, which I just finished. It is a complete slog at the start, with endless boring architectural descriptions. I listened to it in audiobook form which made that section much more pleasant.