## Is mathematics discovered or invented

• 189
Suppose that one we discover intelligent life on another planet, and we find a way to communicate with them (not likely because of speed of light limit, but this is only a thought experiment).

Suppose that they send us what their main results in mathematics, and we see that they have "invented" a different axiomatization for real and complex numbers, but they have a "fundamental theorem of algebra" that is basically the same as ours (every polynomial of degree n on complex numbers has n roots), and they have their Pythagoras' theorem that it the same as ours.

Well, I would say that this would be the "proof" that there is some underlying objective reality for the concept of real and complex numbers and for Pythagoras' theorem.

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?):

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.

Do you agree?
• 2.1k
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.

In theory, if maths axioms were based solely on observation of reality, then our maths should be the same as the alien's maths. 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.

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.

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.

Very much so. Maths is logic is information and information predates everything.

It would take logic to invent logic so logic cannot be invented - it must be a discovery.
• 189
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

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.
But you can prove the fundamental theorem of algebra even using a definition of complex and real numbers not even based on set theory and first order logic (for example type in homotopy type theory).
So, I would say that the theorem corresponds to some more fundamental fact of reality, even if the axiom of infinity could be only one of the many models that can be used to interpret the theorem.

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

Yes, so in my opinion euclidean geometry has an objective underlying reality, even if it doesn't correspond to the physical space-time.
• 1.5k
Logic gives so much systemic and rigorous structure to mathematics that it's quite reasonable for us to argue that mathematical truths are discovered.

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.
• 2.1k
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.

I believe the axiom of infinity does introduce inconsistencies, but that is for another post.

I am a bit old fashioned; I believe an axiom needs to be more than just consistent with the rest of the system. Axioms should chosen because they are inductively very likely to be true. We should have strong reasons for believing in our axioms.

If we allow selection of axioms just on the basis that they do not introduce inconsistencies, our math is likely to diverge from alien's maths - the set of possible axioms is infinite - so we are not guaranteed to choose the same ones as the aliens.

Yes, so in my opinion euclidean geometry has an objective underlying reality, even if it doesn't correspond to the physical space-time.

Reminds me of the theory of forms a bit. Concepts like perfect circles, triangles, euclidean space all seem to exist independently of any particular mind. I think these concepts don't actually have separate existence rather they are deducible from our senses. So we (and the aliens) see approximate circles and triangles in nature and take the idea from that? Ultimately all our information is derived/deduced/induced from our senses.
• 818
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.

If, however, you are talking about the mathematics that we historically developed (and you are likely thinking about Euclidean geometry and 18th-early 20th century algebra, calculus and statistics, as that is what we mostly study at school and what is most widely used in the sciences), that is a very different question. You mention different axiomatizations of numbers, but you don't question the concept of a number itself, because it seems very natural and indispensable (at least to a contemporary person with some education) to think in terms of numbers. You can construct different axiomatizations of numbers in terms of more primitive concepts, such as sets, but the concept of a number is pretty much assumed beforehand: we know what properties we want that entity to have, we have the requirements. Likewise with lines and other geometrical entities.

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. It is difficult to speculate about inhabitants of other planets, but if they are what we usually imagine them to be, that is to say, cognitively quite similar to us and, of course, sharing the same universe with us, then it is not unreasonable to suppose, as you did, that they would converge on the same or very similar concepts of numbers and geometrical objects as we did - and then, of course, they could not fail, as long as they have the mental capacity for it, to prove all the same theorems about those mathematical entities: how could they not if they have presupposed essentially the same properties?
• 189
Concepts like perfect circles, triangles and euclidean space are a description of some properties that a given class of physical objects have in common. At the same way, natural numbers describe the property of sets of physical objects of being put in a one-to-one correspondence.
So, an object is a triangle if it is made of three points connected by three straight lines. I can call a nail "point" and a piece of rope "straight line", if I know that i have to take into account only some of the properties of the physical objects. At the same way, I can call a set of five stones "5" and treat it as a number, if I use only the property of the stones to be distinguishable.
In this way, the definition triangle is "any physical object that can be recognized having three points connected by three straight lines", following an appropriately defined physical experiment.
• 189
I agree with you when you say that
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
. 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.

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

That's exactly my point: how can you decide if something abstract as for example a topological space corresponds to something physical (as in case of natural numbers) or is simply one of the infinite logical games that can be built out of our fantasy?
Well, my opinion is that not all logical games are capable of producing "interesting" results, and that "interesting" results are somehow independent of the logical rules and axioms that you use, but correspond to real characteristics of our universe.
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.
• 189
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
.

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.
• 818
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.

Surely, we don't need the example of another civilization independently "discovering" mathematics to assure ourselves of the "unreasonable effectiveness of mathematics" in describing the natural world (as Eugene Wigner famously put it)? I think our own example provides plenty of evidence of that. 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?
• 4.1k
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 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.
• 3.8k
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.

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.
• 1.5k
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.
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.

For the ugly one you have to have a laudatur in math from the university and knowledge of the distinct field of math to understand what the gibberish is all about. And usually it has not applications.

In a way, a beautiful mathematical object, be it a theorem, proof or whatever, is something that could be given as an example of how interconnected and logical, 'beautiful', mathematics itself is.

I would say something like the Fibonnaci sequence is an example of mathematical beauty: simple and quite useful in various fields.
• 32
If we said mathematicians neither discover or invent but act mathematically in a harmonized way, would this suffice to give it the metaphysical foundation we long to have? If you ask why do we harmonize, I would say because we are similar in out make up and live in the same world.

Or could we simply turn it around, and say the world causes us to react mathematically? No need to talk about inventions and discoveries.
• 1.5k
Or could we simply turn it around, and say the world causes us to react mathematically? No need to talk about inventions and discoveries.
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.

Animal reasoning and extra-terrestial intelligent life may be different from ours, but it is hard to see that there wouldn't be similarities. For example our base-10 number system is quite random, we just have ten fingers and ten toes to easily to count with. Yet that there would not exist some numeral system for finite arithmetic or that there wouldn't be arithmetic would be quite spectacular.
• 2
Do you agree?

Nothing in this world can be invented, Invention has a very different meaning which doesn't relate the human being but we can only discover which is already there.
• 32
I could imagine a life form similar to us but only sees the differences in objects and thus has a primitive concept of “one”. Not sure how successful such a being will be in this universe, but definitely will go about their business differently than us.
• 818
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'?

For a long time there prevailed a sense of a metaphysical or a logical necessity of our mathematical constructs. Pythagoreans, for example, went so far as to put numbers at the center of their metaphysics. 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. But there may be a sense in which the privileged status of certain mathematical structures can be recovered. If so, then when mathematicians describe those structures, it can be said that they are making a discovery, in the same sense in which explorers and scientists - and yes, inventors - make discoveries.

A wheel is a device that is well-suited to a very specific set of constraints: the constraints of physics, scale, local environment, etc. And when something that has such specific objective constraints is produced, we are justified in calling it a discovery. Likewise, evolution discovers adaptations (albeit through a blind process of trial and error) - it does not invent them in an act of pure creativity, because nearly all such inventions are doomed to fail in the face of objective environmental constraints.
• 186
It seems to me that, from the perspective of the absolute, logic was discovered, not invented, and mathematics was invented, not discovered, but from the perspective of the transient part that is man, both logic and mathematics were discovered, not invented.

It is often presumed that one is eternally equal to itself and that therefore the mathematical identity, one plus one equals two, is most assuredly, eternally true, and that therefore, the entire system of mathematics, which follows by necessity from the truthiness of this supposedly absolute identity, is eternally true as well; but does not the equality one equals one beg the question, “one what?” What does the number one refer or point to? Must the number one point to the essence of some particular thing, whether it be conceptual and purely abstract, or concrete and spatially extended in its nature, that actually exists, or can it point to nothing at all and therefore exist eternally in itself apart from anything else as an abstract number which refers to nothing that floats by its lonesome self in a sea of nothingness. I suppose this raises the question then, can an abstract value which retains its identity over time exist apart from time? If it cannot exist apart from time, it cannot exist apart from essence; and if it can hold true apart from time, or memory, how does the number one as a purely abstract value which points to nothing retain its identity? Does the number one necessarily have ontological value in the absolute sense of the word, or does it not? And if one can be a reference to nothing, that is, something which does not possess an essence, yet still remains equal to itself, as such, is not mathematics in the absence of ontology then akin to calculating the number of angels that can dance on the head of a pin and therefore meaningless? Relatively speaking, a number cannot exist apart from the thing or concept in which it represents, so wherefore originated the idea that numbers can exist in themselves apart from at least one other existent thing that is ontologically one in itself? Do not wish to argue that mathematics does not have practical value in the relative sense of the word, but that mathematics has practical value only when the value of one refers to something ontological, that is, something which has actual being, whether it be a physical object or an abstract concept, and that therefore, mathematics should root its foundations, not in the clouds of nothingness, as is currently so, but in being itself in the non-relative sense. 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.

Further, if mathematics has its root in the law of identity 1 = 1, how is it that more complex algebraic identities were abstracted from it, or do they follow by necessity from it and are thus true so long as the law of identity has been true? According to my philosophy, between the law of identity and law of non-contradiction and mathematics and physics, there lies, necessarily, subjectivity, that is, consciousness, for one cannot go from the law of identity and the law of non-contradiction to more complex mathematical identities without the comparison of at least two abstract concepts and an abstraction of a third from them, that is, an intellect.
• 827

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.

As to mathematics in general, I find that none could be possible in the absence of laws of thought. Some laws of thought can be argued to be, at least in part, invented by us. The principle of sufficient reason here comes to mind; this since there are some things that are factual and which are nevertheless arational (i.e., beyond the boundaries of reasoning, as contrasted to the irrational, here strictly meaning “erroneous reasoning”). A primary example of the arational is the very being of being. Yet, in contrast, other laws of thought can arguably only be discovered. The law of identity serves as a good likely example.

I mention this because I then find the question of what aspects of mathematics are discovered v. invented to be in many ways reducible to the question of what laws or thought, if any, are discovered instead of concocted by us.

For instance, if the law of identity is something existentially determinate which is discovered, rather than something only imagined, then it seems to follow that so too can only be discovered the distinction between the following two: the abstraction of an integral whole of quantity—which we represent by the symbol “1”—and the abstraction of an absence of quantity—which we represent by the symbol “0”. These two abstractions of identity then serve as metaphysical limitations to what identity can be. For instance, in the typical process theory of becoming, no given will either be a strict “1” or “0”—for, given that everything is in flux, no given is either a perfectly integral whole nor is it a perfect non-quantity. Nevertheless, here, 1 and 0 yet serve as limiting extremes to what identity can be.

In sum, I therefor assume that 1 and 0—thus understood as symbolic representations for “an integral unitary quantity” and for “non-quantity”—are as essential to any awareness of reality as is the law of identity. The mathematical—and, if I’ve argued it properly enough, metaphysical—notions of 1 and 0 can thereby only be the discovered limiting factors of existence. They cannot be mere fabrications devoid of truth—for they are determinate limits of what can be.

And, in theoretical understandings of mathematics, I fail to comprehend how any mathematics can be accomplished in the complete absence of these two notions which we codify via “1” and “0”.

p.s. 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.

Edit:

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.

You beat me to the punch. :smile:
• 186
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.

I am certainly no mathematician, but my presumption is that both one and zero stand for mathematical waves of a particular frequency which are either in a state of potentiality (i.e. 0), or in a state of actuality (i.e. 1). According to my understanding, there is no such thing as a mathematical wave which is both actualized and not actualized at the same time and in the same respect, 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.
• 827
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).

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.

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.

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.
• 32
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?
• 186
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.

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.

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, so according to my understanding, the law of non-contradiction is contained conceptually as a subset within the the law of identity, meaning that if the law of identity is eternal, so is the law of non-contradiction, and this is because the law of identity (a = a) is identical with the the identity (a = a ≠ -a).

In terms of the difference between physics and mathematics itself, which seems to the basis of physics, and logic the basis of mathematics, their cannot exist waves which are either on or off, without the prior existence of mathematics and therefore logic. According to my understanding, the entirety of mathematics presupposes physics, and also, when a wave is actualized as opposed to not, the law of identity becomes 1 = 1 as opposed to a = a, and is therefore, mathematical as opposed to logical. Have I answered your question? speaking of this stuff is new to me, however, I've thought about the laws of thought, ironically, quite a bit. :)
• 827
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?

Ok, you made me curious.

Unless you’d be one to presume that reality should follow your inventions in all cases without exception, if 1+1=2 as invention were to be discovered to sometimes not apply to reality (say that you’d sometimes experience that 1 and 1 equate to 3), on what grounds would your persist holding fast to this invented mathematics of 1+1=2?
• 827
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,

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.)
• 186
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.)

I wouldn't say that the law of identity applies to nothingness, because the variable a cannot point to something which does not possess an essence, 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)

I wrote a comment of my own about five comments up in which I touched on a few of these issues, maybe not thoroughly enough, but at least in some detail.
• 827
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)

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).

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?
• 186
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?

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? You're really forcing me to understand my own conception of what is and what is not here, and I must thank you, for I've been posting on philosophy groups on facebook for a few years now and there aren't very many seasoned philosophers in those groups, to say the least, so I've never been forced to elaborate in great detail, my conception of reality.
• 32
. . + : = . :

If one invented symbolism “2 + 2 = 4” for the above reality, they may not find much success. However, “2 + 2 = 3” might have some use. My grounds are does the symbolism produce some value in its application to experience.
• 827
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.

I agree with this. :up: Still, it doesn't change the fact that many people find the concept of "Absolute Non-Existence" meaningful. But I see how this could fit in with your model.

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?

To be honest, no, I'm not yet understanding. The concept might need some further fleshing out, though. For instance, why must a "potentiality [...] necessarily beyond space" be a quantity (of one or less) rather than being a complete non-quantity? Also, if the potentiality is a unity that contains multiplicity which, in turn, does not of itself contain a unity, how would this unity-as-potentiality (hence a "1") then be differentiated from a unity that is an actuality (and, hence, also a "1")?

You're really forcing me to understand my own conception of what is and what is not here, [...]

I'm very glad I don't come across as adversarial or some such. Yup, that's what philosophical debates are all about, in the best of times at least. :smile: I myself much prefer the experience of discovering new truths via enquiry over not so discovering.
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