If I understand you correctly, then I think we agree.
Like the cogito ergo sum argument, ascribing axioms to the nature they describe conflates the subjective with the objective. For example, if I write a computer program that runs a simulation which calls for an abstract object or property - for example: one electron, or one topology, or one polygon of a given dimension within the defined scale of the program - every time it renders a thing or moves a thing, and that abstraction exists as a single entry in a single instance of a library in memory, then it’s more accurate to say that part of that thing is the same part as all other instances of the same part in all other things. Now, I’m not saying this is necessarily true. It could quite well not be. But the mere fact that it can’t be disproved casts doubt on the uniqueness of things. It could well be that the very axioms of mathematics which we’ve happened as a species to analyze the universe with are in no way special to the universe, and that there are other axiomatic systems which yield equally valid descriptions.
The question I would then have is whether there is a more fundamental logic that can show different mathematical systems to be logically related. If not, we’d have to face the possibility that different axiomatic mathematical systems describing what we think of as the same natural universe might be fundamentally incompatible, which then casts doubt on our concept of a unified reality. This is in fact what I was getting at months ago in the Simulation Argument thread, that our very understanding of what a simulation is and isn’t might be hopelessly naive…that, on one hand the things we call simulations which we make simple versions of in computers and on paper, and on the other hand the thing we call base reality or physical reality or even just not an artifact, might only be two simple categories in a much larger phase space of heretofore imagined alternatives.
We might eventually have to face the possibility that being finite and limited to subjective knowledge, we can never truly know the whole elephant. But in realizing that the part we’re grasping isn’t the whole elephant, we might at least expand our tool-set(s) for understanding more of the elephant.
They exist as an artificial abstraction. But what is concrete reality? How do you define it and what are the symbols and methods you used to arrive at that definition. The problem is this: Yes, we know the telescope is an artifact we made, but we’re uncertain how much it, and thus we, invent the things we see through the telescope. Our crisis isn’t mistaking the map for the territory. It’s being able to only perceive the map. It’s true this is in part a problem of epistemology. But the thing to understand is that in mathematics we know the map is a map and we can draw other maps that are self-consistent but contradict the first, and we don’t know which if any of the maps most closely describes the territory or even what the territory is or means.
“The nerve of those Zargatchnoi, expecting our votes to be Treasurer of the Galactic Arm Residents Association, when everyone knows that they’re the ones who Uplifted humans.”
All we know for certain is that “in the beginning the Universe was created. This has made a lot of people very angry and been widely regarded as a bad move.”
I did maths up to second year at university in the early seventies. I haven’t used set theory since, except for the Chinese Remainder Theorem, which I needed for a halftoning problem. I remember hating some horrible notation with unbalanced brackets. I am amazed as much came back as it did.
Tangent: I’ve seen it seriously argued by professional philosophers and historians that a decent chunk of Descarte’s philosophy was deliberately bad, on the basis that he was trying to discredit popular positions that he disagreed with. They couldn’t believe that such a talented thinker would push things they viewed as obviously daft (e.g. Cartesian dualism).
I think addition necessarily has something to do with noticing. The universe having any properties beyond merely being there (where “being there” isn’t something we can rightly comprehend) seems to have something to do with noticing. According to some physics experiment I’m familiar with, even things simply existing appears to have something to do with noticing.
If you think about the chaos of base reality, as best we understand it, with fundamental particles existing in probability clouds exerting a handful of forces on one another, and consider that somehow out of that which 1027 or 1028 of those particles are working together to detect other things made of similar numbers of particles to ingest them and somehow continue the functioning of the collective, and that happened by pure chance, the fact that mathematics could have likewise evolved to be good as describing things we observe doesn’t actually seem very impressive, as least it doesn’t to me.
I think mathematics as we use it is a fundamental logic that allows comparison between different sets of axioms. And it’s easy enough to prove that the set of axioms that that would result in systems isomorphic to any mathematical system is infinite, and that at the same time, from a perspective of attempting to model something that really is - that has properties whether we successfully model them or not - these systems are all deficient in different ways.
In mathematics we only care about what can be proven, and one of the things we can prove is that we can’t prove everything. If there is something that is, no model will ever be the whole thing. And with the universe there is also a physical constraint - the model would have the be larger than the universe.
You were talking about considering whether to consider things the same because they are in the same category in a computer simulation. But for everyone but platonists, the 3 I just typed is different from the 3 I just typed. We consider them the same because it’s hard for us to imagine why one three is different from another, but is cosmic radiation corrupts the physical memory device that they are stored on on some server hosting the BB forums, then could matter that one was in one place and the other in another.
We pretend away real differences in the underlying substance of the universe when we try to make mathematics the universe, but the fact that each bit of matter is different from each other bit of matter is why we are alive or dead. Fundamental particles being agitated by energy emissions can give us cancer, it’s not a quibble that can be overlooked. When I wrote down proofs on my exams in university, I was writing the “same proof” as the person next to me, but they were on different pieces of paper written by different ink. I have the “same genes” encoding how to make hemoglobin as they do, but a more precise description of underlying reality is probably that each of us has trillions of little molecules that - if left alone in an arbitrary part of the universe - mean nothing and just are.
I don’t just think it’s possible we can’t know reality, I think it’s obvious we can’t know reality.
I’ve heard people advance arguments like this about other dead philosophers too. It kind of hurts my head. Sometimes there is real evidence that someone was being satirical or trying to make push someone into a revelation by making a false claim, but unless you can give some historical context to explain it you’re just putting them up on a pedestal. There’s no amount of intelligence that lets you escape the time you were born in, you can only really build new good ideas adjacent to the ideas that have already been developed. If you are more than a few years ahead of your time you just seem insane.
And that aside, if we have a book from hundreds of years ago, that’s a better indication that the author was wealthy than intelligent.
I agree completely. But knowing that we don’t directly perceive reality is only a problem if we lazily delude ourselves into presuming that we do directly experience it, that we aren’t essentially map makers. Since human perceptions appear to have always worked along these lines, it seems like popular idealism of a naive sort to say that it is really a problem. I think that the real problem is how people prefer to deny this. It is relatively easy to get a person to stop believing something, but nearly impossible to prevent them from simply choosing to believe something else, rather than to dispense with belief altogether.
I agree, and I agree with @anon50609448 that we probably can’t directly perceive the base reality (or noumena as philosophers would say). But because we can’t directly perceive reality, I think we even have to question what that means. We construct this distinction between phenomena and noumena because we assume there’s a sort of binary difference. Why? Because we’re used to the idea of cause-and-effect, so we assume this extends beyond the limits of our perception. But causality is an empirical observation. Moreover, it’s one that, while it does hold sway in the more fundamental realm of quantum mechanics, it does so in a probabilistic way that’s alien to the deterministic way our minds and senses have evolved to interpret the universe. This to me suggests we should not be especially confident in the actual existence of this base reality. Sure it’s a helpful tool in using our maps, but we should be willing to doubt it.
Now I’m not a mathematician (though theoretical physics has a related problem), but I get the impression that what Wolfram was getting at in the interview is not that we should assume we can understand base reality, but that we have to realize that precisely because we can’t, we must realize our mathematics might not be the only one we can use to do things like study systems and even make useful models of the physical world. Our system of mathematics has been useful, but it’s self-consistency has also caused us to search for mathematical tools that are consistent with that particular system we’ve built on axioms that ultimately derived from how we perceive the world (AKA geometry and arithmetic, and then later on algebra and set theory).
As a physicist, I’m sometimes troubled by the confidence theorists have in the elegance of mathematics. I think it might be giving them a case of confirmation bias…not that they’re approach is wrong (that’s why we have experimental physics, to point the telescope so to speak), but that they could be limiting themselves to a tiny fraction of the possible approaches by their desire for tools and solutions that appeal to their aesthetic senses.
But if this is true, aren’t they using tools outside of mathematics already - that is, their aesthetic sense?
I know that sounds weird and round-about, but when I talk about mathematics having evolved to be useful in describing the universe, I think what I’d add to that is that it’s useful in describing the universe to us. It’s obvious there is an infinite set of alternative axioms and symbols that could be used to add apples, but the vast majority of them would require huge labour to work with. When you look at intentionally obscure coding languages (wikipedia has a list of some) it’s pretty obvious that even if you had been coding for 20 years in INTERCAL or Malbolge you’d switch over to JavaScript in a heartbeat if you realized it was an option.
The inner workings of people’s brains change from generation to generation as they are exposed to different ideas growing up, so it will become possible to consider ideas to radical to consider right now, but I think we need to fall back on our intuition about what’s useful to us now as that is itself a evolved idea that was calibrated to bring us to ideas like mathematics in the first place.
I agree, I just wonder how much that might be because it’s the system we’re exposed to the most and how much it’s innately compatible to how we’ve evolved (that is to ask: how much is the wiring nature or nurture?).
Obviously it seemed “natural” to the ancient Sumerians, Babylonians, Greeks and Indians who laid the foundations of our system, but they were already working within a cultural framework. Just as learning a primary language shapes how one thinks about language generally, I wonder if the the same could be true of mathematical systems.
Also, even if something doesn’t or cannot feel natural to humans, it doesn’t necessarily mean it’s the most effective way for understanding something “outside” of us. There are things that would take many more lines of code in Java that you could do much more efficiently in C++. And while in principle any programming language can be used to achieve the same ends as any other, we shouldn’t assume math is the same. Programming languages work that way because they all follow the same underlying mathematical rules (on classical computers anyway).
Which returns me to the question I raised before of whether there’s a shared logical base for axiomatically different mathematical systems.
Also, I mention classical computers because quantum computers require different mathematics. That’s actually what I do for my bread and butter. I’m basically a glorified translator of algorithms for nonlinear problems. But there are some things a classical computer’s mathematics can’t solve in finite time. I wonder: what is the relationship between different mathematical systems, if any?
Coming from a pure mathematics background I have yet to encounter a system that I would say is a fundamentally different kind of mathematics rather than a different practical application of mathematics. My first post in this thread was talking about how something like “commutativity of addition” might exist in the universe in a pre-thinking-thing state, but fundamentally the problem with the idea of commutativity of addition being in the universe is that even if it is that doesn’t show that the universe has math in it. Commutativity of addition isn’t in mathematics.
Sure, in many sets that we have defined an addition operator over that operator is commutative, but it certainly doesn’t commute when adding strings. “Hello” + " World" != " World" + “Hello”. If I created a mathematical system to describe putting together jigsaw puzzles, I highly doubt addition would be commutative. “That’s not addition” is a No True Scotsman argument. I can call it addition if I want. A professor I once had said, “People sometimes assume commutativity because that’s how they’ve been adding and multiplying their whole lives, but most operations do not commute. Taking off your clothes and getting into the shower are not commutative.”
If there’s anything that’s like a grand unifying theory of mathematics I guess it would be set theory, but even then that’s not really necessary to mathematics. Because of this thread I spent some time thinking about what I think is the very essence of mathematics, and I think I have it: Considering a set of assumptions in a vacuum. That’s really what mathematics is about.
So when Euclid says that you can draw a line between two points, you can’t import any external notions about what a “line” is or what a “point” is (That’s not to say that Euclid himself didn’t import such notions, he probably did, contemporary mathematics is not very old, I’d say it’s younger than the incompleteness theorem). In university we would have heated debates about whether ℕ contained 0. I would say I didn’t care if ℕ contained 0, but if it did then 0 + 0 = 1*. That’s mathematics, not caring what symbol you use to refer to the additive identity, not importing any assumptions from outside.
I think that’s the innovation that has made a system that is so good at helping us with difficult contemporary science problems that require us to look past our pre-conceived notions of how things work (which are increasingly useless for advanced science).
* I actually usually consider ℕ including an additive identity and talk about ℤ+ if I want a set that doesn’t for some crazy reason, this was a joke (Yes, it got laughs)
Yes, sorry, I wasn’t very clear. you can’t translate all algorithms from the math a quantum computer applies to that which a classical computer applies. I didn’t mean to imply they used fundamentally different mathematical systems, but rather that just as you can’t always get there from here in programming one or the other, perhaps you can’t always get here from there in going from one mathematical system to another. It wasn’t the clearest analogy on my part. Anyway, my question is: just as the different mathematical systems of the two classes of computers derive from a common axiomatic system of mathematics, could axiomatic systems of mathematics reasonably be expected to share a deeper common logic?
I like that. And I think it shows how mathematicians think very differently (more abstractly) than scientists. I not sure that ever would have occurred to me anyway. So then my question would be are there rules that can be said to always apply to that processes? And what would we call that logic if not mathematics?
I’ll tell you what I called logic after taking a bunch of mathematics: EASY!
I think mathematicians are better at considering things in isolation than logicians are (I knew plenty of each among my professors). Philosophy is basically 80%-90% noticing other people’s hidden assumptions. In math either a statement is proven or it isn’t. And if it isn’t you don’t have a debate, the proof just doesn’t work.
I was thinking about this too. I mean, classical logic - two truth values and a NAND gate - seems pretty baked into mathematics, but I don’t think you’d find mathematicians especially perturbed if you said, “Oh, I’m using three truth values and this operator…” I think mathematicians would be the least perturbed of all people, really. I wonder what the limits of that are. I’m trying to imagine a system of assigning truth values to statements that would be really different, but I guess it’s too far out of my own internal workings.
It feels like the only think I can think of that that is ultimately just incompatible with approach is a system where context is everything, like (probably) the actual underlying universe.
Aren’t you are confusing haecceity with quiddity here? Sure there are different signifiers and instances of 3. But how could any maths, or sports, or shopping, or anything county work unless there was also an unchanging abstract reality behind the signifiers – at least in the minds of people?
I think that’s because you’re concentrating on the word ‘noticing’ rather than the word ‘mind’ when I suggested that “addition/augmentation […] need not concern itself with a ‘noticing mind’”. That something is bigger than it was can be detected by a machine which is ‘aware’ (yikes - it’s next to impossible to use words without intending to imply intelligence) of states (before, after etc) and relations (more than, equal to, etc). Of course addition needs a ‘detecting mechanism’ to ‘see’ that it has happened. But I don’t think that’s as complex as a noticing mind. I suppose you may then argue that somebody has to be there to notice that the detection has taken place though. But that’s a further (undeniably related) event which might just not happen to happen.
