Is mathematics discovered or invented?

[YellowSubmarine]

...

I don't think that's the reason. I think it actually comes down to the question of why the laws of the physics are the way they are, which happens to be the one thing that one cannot answer as a scientist, because it is a philosophical question. All a scientist can do is demonstrate how these laws are necessary for us to exist and experience the world as we do. Yet we would keep asking the scientists that, and they would keep asking themselves too. And they would try to find simpler, more consistent or more elegant laws of physics just to make it less of an issue, even after we've gotten as far as we can.

However, if the universe is nothing but information, that partly solves the question, or rather it's no longer clear if it still applies. Information operates by some fancy rules that are different from what we have learned to expect from the physical reality. In particular, because of misinformation as a phenomenon, having no information may be equivalent to having all of it in many situations. So it might be that there are no prerequisites for an informational universe to be there, it just would be. And by involving information into the mix, the answer would be no answer.

And from there follows everything crazy – multi-world interpretation of quantum mechanics, other parallel universes – they are all just there, and ‘it's all math!’ None of these are scientific or in the purview of physics, of course.

Being a seemingly simple answer is what makes it tempting – I'm not highly educated in physics, and it still tempts me – half of the time I'm convinced that's the answer, don't know how I would feel if the laws of physics I knew intimately and worked with every day kept reminding me of this. Quantum mechanics seems to beg for a multi-world interpretation (although any interpretation is unscientific and has no bearing on the actual QM), and tends to treat everything as information, and the math looks like nothing my eyes ever saw, so if I worked that every day... Yeah.

(On the other hand, the answer has to be simple, even trivial, because at some point of explaining everything you need stop to avoid following the turtles all the way down. So it's certainly a candidate.)

 

[Crazy Eddie]

I don't think that's the reason. I think it actually comes down to the question of why the laws of the physics are the way they are, which happens to be the one thing that one cannot answer as a scientist, because it is a philosophical question. All a scientist can do is demonstrate how these laws are necessary for us to exist and experience the world as we do. Yet we would keep asking the scientists that, and they would keep asking themselves too. And they would try to find simpler, more consistent or more elegant laws of physics just to make it less of an issue, even after we've gotten as far as we can.

As you say, this is where a lot of junk science comes from IMO. The real answer to that question is, if it the laws of physics weren't the way they are now, they'd simply be some other way. There is no room in physics for "why" because as soon as you start asking that you are immediately over the border into META physics.

However, if the universe is nothing but information, that partly solves the question

It isn't, though. The universe is matter and energy. Information can be attributed to matter and energy and the interactions between them, but they are not the same thing. To suggest that "the universe is information" is like saying "barbecue sauce is a smell." That may be true in a limited context, but if you take that context too far, you just confuse yourself.

Quantum mechanics seems to beg for a multi-world interpretation

Only if you believe in non-deterministic universe.

 

[YellowSubmarine]

It isn't, though. The universe is matter and energy.

Well, the universe is made of matter, which are elementary particles occupying given positions, having given electric charge, colour, and a gazillion other quantum properties I cannot bother to fully understand (they lost me at weak isospin). Some quantum physicists would claim that these observable properties, which do make up information – the particle flavour, position, velocity, quantum states, etc. – are all that there is, therefore the universe is made of quantum information. I don't really mind, as it would mean my knowledge of computer programming and cryptography may be more suitable for answering the philosophical nonsense than their knowledge of physics if they are correct.

Matter is just a word for stuff.

 

[Robert Maxwell]

Mathematics is an innate property of the Universe and is discovered.

The number systems we use to describe mathematics are invented.

There is no mathematics without abstract symbols. Or you are making one of those tedious philosophical arguments that mathematics exists whether we are capable of describing it or not. But that's just not true. The rules of the universe are the rules of the universe. We use mathematics to describe those rules, and our descriptions remain insufficient. The rules of the universe exist whether we can describe them or not, but to claim that is the same thing as "mathematics" is far too anthropocentric.

 

[Crazy Eddie]

Well, the universe is made of matter, which are elementary particles occupying given positions, having given electric charge, colour, and a gazillion other quantum properties I cannot bother to fully understand (they lost me at weak isospin). Some quantum physicists would claim that these observable properties, which do make up information – the particle flavour, position, velocity, quantum states, etc. – are all that there is...

And they're incorrect, because that's what "energy" is. The information they're describing is the STATE of that energy in the force fields between and inside of those particles.

Information is the description of those states when observed or recorded. But a description of a structure is a different thing from the structure itself.

 

[φ of π]

There is no mathematics without abstract symbols. Or you are making one of those tedious philosophical arguments that mathematics exists whether we are capable of describing it or not. But that's just not true. The rules of the universe are the rules of the universe. We use mathematics to describe those rules, and our descriptions remain insufficient. The rules of the universe exist whether we can describe them or not, but to claim that is the same thing as "mathematics" is far too anthropocentric.

So... one stone and one stone only makes two stones if there's someone there to count them?

 

[YellowSubmarine]

And they're incorrect, because that's what "energy" is.

Claiming that a philosophical interpretation of what matter intrinsicly is.. to be incorrect (or claiming it to be the correct one, for that matter), is problematic, given that the question is unanswerable. We only observe matter and its properties in the form of information, what it actually is beyond that is unclear and probably unknowable. It's also irrelevant for any physics we observe, as it bears no consequence on the real world.

 

[Robert Maxwell]

So... one stone and one stone only makes two stones if there's someone there to count them?

I assume you're being flippant, but... yes. First, distinguishing what constitutes a "stone" is subjective. The concept of a stone is constructed. If you take a stone and break it in half, is it now two stones? If it was a stone before and after being broken it is still stone(s), what makes it a stone in the first place, and how do you reconcile that its quantity is time-dependent? If you go back far enough, the stone didn't exist--it was part of the molten mass comprising the primordial Earth. Go back further, and it's interstellar debris--many stones, perhaps. And someday in the future, it will be absorbed into the growing sun and cease being a stone. Its stoneness persists only in the fact that we, here and now, claim it is a stone, and a single stone, up to the point where we smash it (or the universe does it for us).

Quantities are useful abstractions for humans, and our brains are quite adept at dealing in quantities, but don't mistake that for being the universe, in and of itself.

I am not saying that the universe has no innate properties which we, as humans, discover. What I am saying is that the ways in which we represent these discoveries--such as through mathematics--are invented. As the saying goes, the map is not the territory. And none would seriously argue that cartography is an innate property of the universe, I reckon.

 

[Asbo Zaprudder]

So... one stone and one stone only makes two stones if there's someone there to count them?

Cardinal, ordinal and nominal numbers were invented by man although some claim that other animals can count (cardinal) although I'm not aware of those animals using ordinal (sorting) or nominal (naming) numbers. I can't disprove that Platonic forms exist for the concepts "two", "second", or "object I choose to identify as number 2". However, with Platonic Forms, one does run into the problem of where to draw the line (as recognised by Plato himself). Is there a Platonic Form for just a receptacle that can hold a liquid, or is there one for each type of paper receptacle intended for Starbucks, Costa Coffee, etc? Is the tableness of a table similar to the chairness of a chair, which also usually has four legs? Do they derive from the same or different Forms? I would maintain that it's only in our minds that numeric concepts exist although our brains might be hard-wired to develop such concepts if they were a survival advantage to our ancestors. How many lions? Oh shit, run away. How many more gazelles that way? Great, let's go hunt those.

Principia Mathematica was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that Principia Mathematica, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.
...
Famously, several hundred pages are required in Principia Mathematica to prove the validity of the proposition 1+1=2.

https://en.wikipedia.org/wiki/Principia_Mathematica

But, again, that's several hundred pages of symbols ultimately derived from the mind of man.

 

[φ of π]

What about physical formulae such as Newton's 2nd Law (Force=mass*acceleration) or the inverse square law that describes gravity's strength decreasing, or the intensity of emitted light decreasing, with the distance from the object?

We Humans may use base-10 numbers and symbols to describe those laws, but they are an inherent part of reality whether we use numbers and symbols to describe them or not.

That's what I mean when I say that mathematics is discovered, as it already exists.

 

[Asbo Zaprudder]

Essentially, those laws are correlations that we deduce by taking measurements. Think of it as an advanced form of dimensional analysis.

https://www.trekbbs.com/threads/is-mathematics-discovered-or-invented.286883/#post-11963950

https://wp.optics.arizona.edu/rfrieden/fisher-information/

 

[publiusr]

It has been postulated that all of Physics is actually based on observer-dependent metric correlation

The horror story "Details" by China Miéville comes to mind.

...by some incredible chance, their version Star Trek describes us in frightening detail. Their laws of physics are so crappy that inter-continental travel and powered flight are incredibly impractical. Their over-the-top mad science fantasy involves aeroplanes and ships travelling between continents in days and hours, and laughable magic like radiowaves. Of course, there would be superfans who go to extreme lengths by dissecting that insane fictional tech in such great detail that you could put it together in a math model.’

Whoever wrote that version deserves a real punch in the face. Great Bird indeed.

 

[Crazy Eddie]

Claiming that a philosophical interpretation of what matter intrinsicly is.. to be incorrect (or claiming it to be the correct one, for that matter), is problematic, given that the question is unanswerable.

I don't see how.

You mess up your ankle doing handsprings in the back yard. Two doctors come and look at your ankle. One of them says "That feeling in your ankle is just information. Your brain is a wonderful computing device, and with enough concentration, you can chose how you interpret that information and how your body reacts to it." The second doctor says "Your ankle is broken. You need to go to a hospital."

You can wax philosophical about a broken ankle all you want, but there is a fine line between "philosophy" and "bullshit" and quantum physicists flirt with that line far too often.

We only observe matter and its properties in the form of information, what it actually is beyond that is unclear and probably unknowable.

Obviously. But to take from that "information is all there is" is, while it works from a philosophical sense, is BEYOND silly when it comes to actually describing the universe. To conflate the observation of a thing with the thing itself is the kind of mistake that, frankly, physicists should be smart enough not to make.

 

[YellowSubmarine]

I don't even see how that analogy makes any sense. In your analogy, you actually have evidence that the observation is consistent with a broken ankle.* In physics, even after you've done a very thorough observation, the most specific thing that you can have evidence for is that there a bunch of particles in a certain quantum state, which you don't even know for the most part. Qualifying what's the nature of those particles beyond that would be the silly thing to do. And you certainly have the option that there isn't anything more to it.**

* Besides, you got it wrong, both doctors told me it was broken last month, and x-ray still decided to disagree with both of them.
** Just like there wasn't with my leg.

 

[Crazy Eddie]

I don't even see how that analogy makes any sense. In your analogy, you actually have evidence that the observation is consistent with a broken ankle.* In physics, even after you've done a very thorough observation, the most specific thing that you can have evidence for is that there a bunch of particles in a certain quantum state, which you don't even know for the most part.

And what you DO know is what you can say. The information you have is the information you can record and express. That was the point of the analogy: the first doctor doesn't know the ankle is broken and is just throwing out some homeopathic fluff in order to make his lack of information sound more meaningful than it really is. The second doctor is making a prediction with a testable hypothesis: the only way to confirm whether or not the ankle is actually broken (as opposed to dislocated or sprained) or how badly it is broken and what to do about it, is if he goes to the hospital and gets an x-ray.

A complete lack of data is a perfectly valid data set. If nothing else, it gives you the parameters of what your questions are supposed to be and what you should be looking for. But extrapolating your lack of data into a positive statement is something entirely different.

Qualifying what's the nature of those particles beyond that would be the silly thing to do. And you certainly have the option that there isn't anything more to it.

Yes, it's certainly an OPTION, just like homeopathy is an option for treating a severe injury to the ankle. You can basically take the data that's right in front of you and deny that it means anything at all and re-interpret it to fit any hare-brained philosophical position you prefer to believe.

But "there is nothing there but information" is not something you can justify scientifically. It's a solipsist position that denies the existence of objective reality outside of the perception of the observer. While that is a perfectly acceptable form of existential navel gazing (e.g. "What if the world and everyone and everything in it is an illusion and I'm really the only one in it and this was all created just to trick me into believing a false reality and I'll never know what the truth really is?") that kind of shit has NO BUSINESS working its way into an actual physical hypothesis. Physics itself is the science of understanding the behavior and underlying geometry of the universe; physical laws can be incorrect, and physicists can make bad guesses from incorrect data sets. What they CAN'T do is deny that the world they're describing even exists at all, because that is not a hypothesis that is in any way testable, nor is it consistent with the underlying assumptions about physics as a discipline.

tl;dr: A doctor who tells his patient "Injuries are all in your head, you can heal yourself if you just convince yourself you are healed" is a bullshit doctor.
For the same reason, the quantum physicist who says to a journalist "The universe is just information and nothing is real, we just think it is because we're observing it" is bullshitting... well, EVERYONE.

 

[YellowSubmarine]

It's a solipsist position that denies the existence of objective reality outside of the perception of the observer.

It does nothing of the sort.

 

[Mudd]

Discovered.

 

[CorporalCaptain]

Discovered.

Hear, hear.