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.
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.