Do wheels have 'objective reality'? — StreetlightX
The question is, how far can we take that conclusion? When we develop mathematical theories and construct mathematical models to explain the regularities in our observations, do we thereby discover some objective truth about nature? — SophistiCat
But would this argument have the same force if applied to, say, wheels? Would we be surprised - or not - that aliens also have wheels? And would this mean that wheels (the quintessential 'invention') are therefore discovered? Do wheels have 'objective reality'?
Or imagine that aliens have cars - or at least, transport vehicles with four wheels. Would this mean that cars have objective reality? Or would it be that four points of contact with the ground works really nicely for stability? (more stable than 3, less unnecessary than 5 - are 4 wheeled cars an objective truth?) And that circular structures are good for things that move? Could it be that math is as it is for similar reasons? All of this is not to 'take a side' in the invented/discovered debate, but only to point out that the 'argument from aliens' is not a particularity strong one. At least, not without a whole bunch of other qualifications. — StreetlightX
The wheel example is perfect for the occasion. It informs us that the invention-discovery distinction is not as yet clear to us. I mean we're misapplying the words ''invention'' to wheels rather than that the aliens ''discovering'' math is wrong.
I think math is both an invention and a discovery. As Sophisticat pointed out we create mathematical worlds using axioms of our choice. These remain in the realm of invention until it finds application in the world after which we see it as a discovery. — TheMadFool
Another way of saying that would be that it's possible for an invention to have a twin in the world. Before we find the twin it's an invention but after we find it it's a discovery. A better example would be, history seems replete with examples, I thinking of a particular theory for the first time (invention) only to find out later that it's an older theory (discovery) — TheMadFool
Math is like that. It's not entirely a discovery because some math have no application (as of yet) and it's not all invention because some math have real-world applications. — TheMadFool
Well, that's not exactly what I had in mind saying that "the fact that two independent civilizations "invent" the same mathematical theorem is a proof that the theorem has an underlying objective reality". I think that the "underlying reality" of the theorem is there even if it were not "invented" by anybody. But the fact that is "invented" at the same way by many independent civilizations would be a proof of the fact that it is not merely one of the infinite combinations of logical symbols that can be built with the formal "logic game". — Mephist
I do not think that mathematical entities exist independently of the mind. That said, I don't doubt that the relational structures embedded in mathematical theories mirror the vastly more complex relational structures existing "out there" in mind-independent reality. — Aaron R
Bertrand Russell said that 'physics is mathematical not because we know so much about the physical world, but because we know so little — Wayfarer
Perhaps it should be noted that any computer follows algorithms in a specific way (referred typically as the program it runs), which makes the whole thing quite mathematical.But could there in his world still be "laws of mathematics" and mathematical theorems? I think the answer is yes! For example number theory is based only on the fact that natural numbers and logical rules are "constructible", and I that is based on a very minimal set of requirements that the "physical universe" must have. — Mephist
I would argue that a lot of things that we take as important yet problematic are indeed mathematical, but simply not computable. Even the patternless are still mathematical.So, I think that there is some set of "interesting" mathematical constructions that are different from pure logical combinatorial games in some concrete sense, and are not really related to our particular laws of physics. — Mephist
Perhaps it should be noted that any computer follows algorithms in a specific way (referred typically as the program it runs), which makes the whole thing quite mathematical. — ssu
For me, if math is a discovery then it must exist in the world ouside of our minds and that, to me, points to the mathematical laws of nature (science). — TheMadFool
This is the point: it's not necessary a "real" world to represent math, but anything capable of following a set of rules (including a "virtual" world)Without a connection to the real world math would only exist in our minds; making it an invention. — TheMadFool
But why did we choose numbers and geometrical objects for our mathematical exploration? Stepping back to an abstract remote once again, they are nothing but mathematical constructs - a drop in an infinite sea of such constructs. There is no a priori reason to favor those concepts over any other. So the reason will not be found in the abstract enterprise of mathematics, but in the world that we inhabit, and perhaps in the contingencies of our cognitive and cultural evolution. — SophistiCat
So my idea is that the axiomatization of mathematical ideas is invented, but our axiomatizations are based on some underlying objective facts of nature that are discovered. And the distinction of what is real from what is invented could be based on a definition of this kind (is it possible to make this definition more precise?) — Mephist
The axiomatization of mathematical ideas is invented, but our axiomatizations are based on some many underlying objective facts observations of nature that are discovered — Pattern-chaser
Isn't everything itself a perfect model of itself?Maybe there are even parts of nature that don't follow any map, but the ones that follow the map are the ones that we may be able to understand. — Mephist
That's the whole point! It's genuinely defining the limits of computable math. What here is important is to understand just how basic patterns are for ordinary mathematics.In my opinion we don't call mathematics what is patternless. — Mephist
Think so?I would argue that nobody would consider a mathematical paper that express patternless theorems. — Mephist
When we don't have a pattern, we can't extrapolate, calculate or do the other usual mathematical stuff. Yet of course something not having a pattern is still logical and still part of mathematics — ssu
Same thing with immeasurability or non-measurability. Take for example the non-measurable sets like the Vitali set — ssu
But this topic asks whether maths is invented or discovered. To believe that maths was discovered is to mistake the map for the territory. For maths is a mapping tool. It helps us understand the universe, in some ways, and that is admirable. But it wasn't discovered, it was invented, just like real, literal maps made of printed paper. There's no shame in that. :smile: — Pattern-chaser
Well, my idea is that there is a "map" that exists in some kind of Platonic world that is taken as a model by nature. Parts of this map are taken as a blueprint from nature to build real things and even human beings. The real things are not perfect as the map, but tend to resemble to them in a high degree. Maybe there are even parts of nature that don't follow any map, but the ones that follow the map are the ones that we may be able to understand. — Mephist
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