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. — Mephist
A mathematical concept is discovered (and then based on an underlying objective reality) if the same concept is present in the mathematics of other intelligent civilizations that evolved independently from ours. — Mephist
But our maths axioms are not all based on reality (axiom of infinity for example) so I think certain parts of maths diverge from reality. — Devans99
It is interesting to note that according to relativity, euclidian geometry diverges from reality. But it is a useful approximation of reality and one that any aliens would no doubt have in their mathematical canon. — Devans99
From the point of view of mathematics, the only relevant thing is that the axioms that we invented are not inconsistent (i.e. not contradictory: they are satisfiable in some model). If the axiom of infinity is not inconsistent, there should be some model in which it is true; so in this model the axiom doesn't diverge from reality. — Mephist
Yes, so in my opinion euclidean geometry has an objective underlying reality, even if it doesn't correspond to the physical space-time. — Mephist
. So, I think that the concepts of numbers and geometrical objects are in some sense related to the physics of our universe, and not simply abstract logical constructions that can be included in a non contradictory theory.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 — SophistiCat
You should be careful about how you ask this question. What mathematics are we talking about? Viewed from the most general perspective, mathematics is a logical game. You set up some rules and then you use those rules to construct abstract structures and prove theorems about them. There is an infinite variability of such games; the enterprise as such does not dictate to you which rules you should select and what you should construct from them, so there is nothing preexisting for you to discover. — SophistiCat
.The given proofs or the mathematical approach taken to establish a mathematical proof can be quite subjective and rely on the person doing the proof and what he or she has been interested in, yet mathematics as a whole is such a beautiful system that the truths aren't just inventions — ssu
So, if an intelligent culture completely independent from ours happens to create the same concepts out of the infinite quantity of possible logical games, that would be a strong indication that there is some meaning in these concepts that is not related to logical games. — Mephist
The 'argument from aliens' ('could aliens have the same math as us?') always struck me as interesting because it's so ambiguous, despite it seeming not to be. The force of it, I assume, comes from its seeming to shore up the 'discovered' position, ('if something so different as an alien uses the same math, this must mean math is "objective"'!). 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 means that cars have objective reality? Or would it be that four points of contact with the ground works really nicely for stability? 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. — StreetlightX
Usually the most beautiful mathematical object (or theorem, proof etc.) is the most simple and the most applicable to various fields of mathematics. In other words, it has equivalent findings in other forms.I agree. But do you think is possible to give a concrete meaning (or measure) to what it means for a theorem to be "beautiful"?
I mean: if a large group of people is able to distinguish a beautiful mathematical theory from an ugly one, probably there exists a measure of "beautifulness" independently from the person that judges. — Mephist
The use of logic makes mathematics as it is. I think it is reasonable to say that logic gives us the mirror how we make sense of the World around us.Or could we simply turn it around, and say the world causes us to react mathematically? No need to talk about inventions and discoveries. — Richard B
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'? — StreetlightX
Closer to our time, logicists hoped to give traditional mathematics an a priori foundation. Recently though these notions have come under attack and have been significantly weakened if not altogether defeated. — SophistiCat
Essentially, if there is not a field of mathematics which concerns mathematics as it relates to ontology, there should be, because without ontology, mathematics is meaningless. — TheGreatArcanum
While criticisms are of course anticipated wherever warranted, I mostly mentioned this perspective because I’m curious to see if anyone knowledgeable of theoretical mathematics knows of any such maths that are fully independent of the notions of 1 and/or 0. — javra
I am certainly no mathematician, but my presumption is that both one and zero stand for mathematical waves of a particular frequency that are either in a state of potentiality (i.e. 0), or in a state of actuality (i.e. 1). — TheGreatArcanum
so the law of non-contradiction extends its reach down into the mircocosm and beyond into the omnipresent field of non-locality which precedes and contains all waves and therefore, all actualized things in relative space and time. — TheGreatArcanum
Isn't this confounding some mathematical models of physics with mathematics per se? For one example, we could address one potentiality as contrasted with two potentialities. — javra
I'm one to support this perspective. I didn't mention the LNC due to the pesky modern notion of dialetheism, which states that the LNC is not a universal law/principle. And its rather difficult to disprove. But yes, when it comes down to it, I agree with your quoted stance. — javra
I have invented the following symbolism “1 + 1 = 2” and I discover that it has many applications in life. What else is there say? Do we add anything of value to say “and by the way there are these eternal things out there that correspond to these symbols” If one said “We have proven they do not exist” What am I suppose do? Give up Mathematic? — Richard B
I suppose that mathematics has its first appearance in the Law of Identity, not a = a, but 1 = 1, and that 1 points to something which has an ontological value, that is, an essence, and an essence which is equal to itself and not equal to its antithesis so long as it exists, — TheGreatArcanum
How then would you make sense of the law of identity specifying that "nothingness" = "nothingness"? This where "nothingness" is defined as absence of essence. It's still a = a, but it no longer seems to be 1 = 1 by the standard you've just provided.
(btw, non-quantity can be givens other than nothingness; examples can include those of Nirvana. And I grant that such latter examples do hold essence. But this is likely a very different topic.) — javra
that is to say that something with an essence, or rather, the potential to point to an essence, which is in itself, an essence, cannot point to something which does not possess an essence (i.e. nothingness) — TheGreatArcanum
But then why does the symbol of "nothingness" as a word point to something that we find meaningful, i.e. something that humans deem to hold an ontological value? Nothingness might be a false concept, but it is yet meaningful conceptually (rather than pure gibberish).
this is a very good question, yes, in the relative sense, 'nothingness' points to something with an essence, albeit a purely abstract and conceptual essence which exists in the realms of abstraction in relation to other things; thus we must distinguish between non-existence in the relative sense, which exists as an abstract concept in relation to other existent things and concepts, and Non-Existence in the absolute sense which cannot exist in relation to Existence, for therein would lie a contradiction, that being the co-existence of both Absolute Non-Existence and Absolute Existence, and if Non-Existence exists in relation to existence, and existence is born out of and contained within Non-Existence, there lies another contradiction, that being the the fact that something Non-Existent cannot possess the potential to contain something existent within itself, for otherwise it would have an Essence and thereby be Existent as opposed to Non-Existent.
As to the symbol of "0" representing potentiality, how again can we then go about saying there is one potentiality rather than two, or none? — javra
this is a very good question, yes, in the relative sense, 'nothingness' points to something with an essence, albeit a purely abstract and conceptual essence which exists in the realms of abstraction in relation to other things; thus we must distinguish between non-existence in the relative sense, which exists as an abstract concept in relation to other existent things and concepts, and Non-Existence in the absolute sense which cannot exist in relation to Existence, for therein would lie a contradiction, that being the co-existence of both Absolute Non-Existence and Absolute Existence, and if Non-Existence exists in relation to existence, and existence is born out of and contained within Non-Existence, there lies another contradiction, that being the the fact that something Non-Existent cannot possess the potential to contain something existent within itself, for otherwise it would have an Essence and thereby be Existent as opposed to Non-Existent. — javra
since that potentiality is necessarily beyond space, it cannot have a quantity more than one. Of course, within itself, my varying concepts can exist in the abstract sense of the word, but they are not mutually exclusive in relation to the whole; therefore, the non-local substratum of potentiality is a unity which contains multiplicity within itself, but not a multiplicity which contains a unity within itself, if that make sense? — TheGreatArcanum
You're really forcing me to understand my own conception of what is and what is not here, [...] — TheGreatArcanum
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