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.)
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.