Is mathematics discovered or invented
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?
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?
Comments (68)
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.
Quoting Mephist
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.
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.
Yes, so in my opinion euclidean geometry has an objective underlying reality, even if it doesn't correspond to the physical space-time.
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.
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.
Quoting Mephist
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.
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?
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
Quoting SophistiCat
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:
Quoting TheGreatArcanum
You beat me to the punch. :smile:
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.
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.
Quoting TheGreatArcanum
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.
Quoting javra
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. :)
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?
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.
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?
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.
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.
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.
Quoting TheGreatArcanum
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")?
Quoting TheGreatArcanum
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.
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. Regarding aliens, it would not surprise me in the least if an intelligent alien culture with a significantly divergent anatomy and physiology developed a mathematics that is all but incomprehensible to us (and vice versa).
The form of a wheel has objective utility. Mathematics is an elaboration of counting, and counting also has an objective utility. It is also an objective fact that things are countable. It seems reasonable to presume that any alien mathematics would necessarily, if it is to qualify as mathematics at all, be based on counting.
I think the answer is YES, and I think there should be an objective way to distinguish if the regularities are due to the way we built our axioms or are objectively laws of nature
The wheel is an invention related to the discovery of some physical facts of nature: the rolling friction of a round body is much smaller that the creeping friction of other shapes. Moreover, the center of a circle moves slower than the border. So, I can fix an axis to the center and use it to transport heavy objects with less force. Similar explanations can be given for most of the other technological inventions: they are related to the discovery of what we could call "laws of nature".
So, to answer your question, wheels (or cars) are a human invention (if we call "invention" the creation of an object that didn't exist before), but the laws of nature on which they are based are "an objective truth" that probably would be discovered by other intelligent beings and used to create machines similar to ours.
I think that we could use quite simple unambiguous definitions of invention and discovery:
-- "invention" is the creation of something that didn't exist before: by "exist" I mean of course that "had not been built", not that did not exist as possibility. For example, a new novel is an "invention", because that novel was not part of the world before being written, even if, of course, every possible novel exists as a possibility, because it's only a long string of characters.
-- "discovery" is the observation of something that is not evident at first sight, or that seems to be "surprising" or "not normal". This is of course a more problematic definition: something can be surprising for some people and normal for others. But in practice it's easy to agree on which facts of nature facts of nature could be qualified as "discoveries". Some examples:
- when Galileo Galilei observed that free falling bodies follow a law of squared times nobody expected a such simple regularity in nature.
- when you see Pythagoras' theorem without being told the demonstration, it seems a strange coincidence that the sum of two squares equals the other square
- when you see Maxwell's equations, it seems very strange that such simple symmetric equations describe so many facts of nature
- when you see the fundamental theorem of algebra, it seems to be a surprising coincidence that all polynomials of degree n have always n solutions
( I could continue with many more obvious examples, but I think I gave you the idea.. )
So, returning to the creation of mathematical worlds using axioms of our choice:
you can create a mountain of mathematical sentences made with casual axioms (a computer can do it even easily and very quickly), but it would be very improbable that something "interesting", or surprising would come out of them: at the same way as you can automatically create syntactically correct novels, or randomly built machines that are useful for nothing.
I think that axioms are like english grammar: they are the syntax of a language, but the meaning of the novel is something more than just a list of grammatically correct sentences.
You're basically saying that the difference between invention and discovery is that the former comes into existence while the latter always existed. I agree with this.
What I find relevant to your question is that nothing precludes a correspondence/match between invention and discovery. 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).
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.
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". Of course, we don't have an independent mathematics built by aliens to compare with, so we cannot be sure which parts of our mathematics (if there are) are merely logical games with no other underlying objective meaning. Or maybe there exist some way to "measure" the importance of theorems, but we didn't discover it yet.. ( see https://thephilosophyforum.com/discussion/5789/is-it-possible-to-define-a-measure-how-interesting-is-a-theorem ).
Quoting TheMadFool
I think that the fact that there is no application it's not a good clue of the fact that there is no objective meaning
Are you confining your argument to mathematics alone or are you also including the mathematical nature of the laws of nature?
There's a difference and it matters.
If you're only talking about math then your argument seems a bit weak because it's not impossible that many or all intelligent life construct identical or similar mathematical worlds. I know many examples where people thought they hit upon a "new" idea, only to discover that it was an old one. Coincidences do occur. Of course, using mathematical probability, we can deduce that parallel development of identical/similar mathematical worlds among different intelligent life is suspicious to say the least. Nevertheless, remember we share the same universe and are exposed to the same elements that stimulate quantitative (math) thinking. It shouldn't be a surprise that we created identical/similar math. In fact it is to be expected.
On the other hand if you say that the laws of nature are universal and mathematical then the argument is stronger since it implies that our universe has a mathematical structure and so we must discover it and not invent. Of course we have to first invent the math and then check if it applies to the world outside. I'd like to refer you to Eugene Wigner's 1960 article The unreasonable effectiveness of math in the natural sciences
Scientific method relies on the ability to capture just those attributes of subjects in such a way as to be able to make quantitative predictions about them. In other words, if you can represent something mathematically, then you can use mathematics to make predictions about it. The greater the amenability of the subject to mathematical description, the more accurate the prediction can be: hence the description of physics as the paradigm of an 'exact science'. Bertrand Russell said that 'physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.' But nevertheless, within that domain, the ability to apply mathematical logic to all manner of real objects yields practically all of the power of scientific method. In other words, what can be expressed in quantitative terms can also be made subject to mathematical analysis and, so, to prediction and control. It becomes computable. That is of the essence of the so-called 'universal science' envisaged on the basis of Cartesian algebraic geometry.
But this also challenges the dichotomy of mathematics being 'in the mind' and the world 'out there'. Things that can be quantified to conform to mathematical predictions in the same way that they conform to logic. We know by mathematics and logic the laws and axioms which are visible to thought itself - Frege's 'laws of thought' - and requiring no empirical validation, on account of them being logically necessary; they're not 'out there'. But through this quantitative method, the certainty availed by logical prediction can be applied to collections of all kinds, not just to empirical particulars , and with mathematical certainty. It's the universal applicability of these procedures to practically any subject which opens access to domains of possibility forever out of reach of an intelligence incapable of counting (whence, Wigner's 'unreasonable efficacy').
I think that in some fundamental respect, real numbers are discovered rather than being invented, but that they're neither in the mind, nor in the world; they pertain to a logical domain which transcends that dichotomy; they are real a priori to that division; but that having the ability to perceive them also bestows the ability to invent other like kinds of ideas; so the distinction 'discovered' or 'invented' can't be hard and fast.
:cheer: :up: Maslow's hammer
suppose that our alien was not a real alien, but a character of some computer game, and he discovers that fact (like in film "Nirvana", one of my favorite ones: https://en.wikipedia.org/wiki/Nirvana_(film)). Obviously, in his world there are no real "laws of physics", because he knows that the author of the game, that has full control of the world where he lives, could make happen whatever he wants: objects or persons can disappear, or move at instant speed, and the whole universe that he sees is only a simulation. So he knows that in his world there are no "laws of physics". 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.
So, with this definition of "laws of mathematics", I believe that it would still be possible to reconstruct most part of what we call mathematics today. 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.
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.
Quoting 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.
What is the evidence?
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). That the laws of nature can be described with numbers and geometry indicates the universe has a mathematical structure. This is as real,thus discovered, as math can ever be.
I think you disagree with what I said above. You made an argument in your last post about the film nirvana and how a person in a simulation could still discover math despite the laws of nature in that simulation being an illusion. Doesn't your argument actually refute your position which, I think, is that math is a discovery? Without a connection to the real world math would only exist in our minds; making it an invention.
Personally speaking, I think math is both an invention and a discovery. I don't think math is a complete discovery because there are many mathematical objects that have no real world counterparts i.e. they exist only in the mind. However, some mathematical objects have real world application i.e. an actual phenomenon matches a given mathematical object.
Let M ={all math objects}
P ={phenomena in the real world}
The intersection of M and P is not the null set. However M - P too is not the null set. Math is both an invention and a discovery.
Yes, but even if the algorithms were subject to any arbitrary changes by the author of the game, he would still be able to prove the same theorems of our mathematics, if he is able to use the objects of his world to build a model of the theory (probably made of symbols), and then verify that these objects have a given set of properties following the rules of logic. At the end, it would be enough to be able to define an algorithm (made of a given set of rules), that behaves always at the same way when you run it with the same input. If in your world there is no way to define how to perform addition such that the sum of the same two numbers gives always the same result, than mathematics for you makes no sense.
I'll give you an example: the game of chess exists "ouside of our minds" as list of possible "positions" and a list of allowed "moves" that the players can perform to pass from a given "position" to the next one. It doesn't matter what physical model you use to represent the positions (usually the chessboard and the pieces) and in what form you write down the set of rules (as soon as you are able use them to decide in an deterministic way if a given move is allowed or not).
So, in whatever "thing" that allows to distinguish objects from one another and to build rules that can be followed in a deterministic way, the game of chess exists.
Our mind is one of those "things" that allows to represent the game, so game of chess can exist in our mind. Our physical world is another "thing" that allows to represent the game. And there are plenty of "things" in which the game can be represented.
Now, the point is that in all of these "things" the game of chess works exactly at the same way. So, if somebody proves one day that there exists a winning strategy for the white player, this theorem is not about our mind or our world, but is rather about all "things" that have the ability do distinguish objects from one another and to follow deterministic rules.
If there were nothing in the universe with these characteristics, then both the game of chess and even mathematics did not exist. But since there are other "things", except our brain, that have these characteristics, both the game of chess and mathematics exist independently of our minds.
Quoting 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)
This is a post from 5 days ago, but it's an interesting subject so I'll reply to it now.
I think that we could find the reason to favor numbers and geometrical objects over the infinite sea of other possible mathematical constructs "in the abstract enterprise of mathematics", without looking at the world that we inhabit. For example, we could find a mathematical function that takes as an input the formulation of a mathematical theory in some formal language and returns a positive number that is a measure of how "interesting" is that theory. The function should be made in such a way that when it's given as input mathematical theories generated by putting together axioms and rules at random, the return value is very low. Instead, if t's given as input mathematical theories that we judge as interesting, the return value is much bigger. This would be a judgment independent of the physical world, "internal" to mathematics itself. I think that it wouldn't be difficult to build such a function with one of the technologies used today for pattern recognition algorithms, such as neural networks. The problem is that to teach the neural network which theories to recognize, you have put them inside yourself, so in reality it would be only a memory of the theories that we judged as interesting. But there is an objective way to decide if the function is only a memory of the things that we just know, or something more: the quantity of information necessary to describe the function should be much smaller than the quantity of information required to write the theories that it recognizes.
Well, in my opinion the theories that we find interesting differ from the others for the fact that we can "encode" a great quantity of apparently unrelated facts from a very small quantity of rules and axioms. In other worlds, they are full of symmetries, and that's basically the reason why we find them "interesting".
Well, in reality I am not convinced that the thing is so simple as I described it. Probably that function would be impossible to calculate even if possible to define, or would not agree completely with what we judge "interesting", but I think that something like this will be discovered one day in mathematics.
The world and our mind have forms that are similar to interesting mathematical objects because these forms are in some way special, and the laws of physics favorite the development these forms (i.e. forms with an high degree of symmetry) respect to the others.
No, I think it's already just a little too precise. I think it should be something like this:
The axiomatization of mathematical ideas is invented, but our axiomatizations are based on [s]some[/s] many underlying [s]objective facts[/s] observations of nature that are discovered.
I believe it's not only from observations of nature that mathematics takes inspiration.
An obvious example: Mandelbrot set is not present in nature, but it's an interesting object. Group theory has started from the solution of polynomial equations, and only after was discovered to be important in physics. Maybe one of the most surprising is Riemanian geometry: invented as a pure abstraction, and then discovered to be exactly what's needed to describe gravitation. Often is the other way around: mathematicians discover some interesting structures that you can build out of pure axiomatic theories, and then it comes out that they are exactly what's needed for physics.
Isn't everything itself a perfect model of itself?
Following a map is somewhat difficult metaphor to understand. You see if there is a pattern, then we can extrapolate from the pattern. Yet something can be patternless, which still has a perfect model of itself and that is itself.
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.
Quoting Mephist
Think so?
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. Same thing with immeasurability or non-measurability. Take for example the non-measurable sets like the Vitali set.
Yes, what I meant is that some parts of mathematics are "interesting" and some are not. And I think this distinction can be made internally to mathematics itself, without looking at nature ( see my other topic https://thephilosophyforum.com/discussion/5789/is-it-possible-to-define-a-measure-how-interesting-is-a-theorem ). So theorems, or rather theories, that have a high value of the "interest" function ( the ones that have a pattern ) are discovered, because the value of the "interest" function for that theory is defined for all theories even if we don't know them. On the other hand, 10 + 10 = 20 is of course logical and true, but there's nothing in it that makes it "interesting"
Quoting ssu
I believe non measurable sets are interesting as part of topology, but they are not a model for the physical space. For the physical space we should use a model where all functions are continuous, so that the Banach-Tarski theorem is false and all objects have a non-zero measure (but possibly an infinitesimal one), such as for example in (https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis), or any other simpler axiomatization that has these characteristics. If the axiomatizations are not equivalent, which one of them is correct is a matter of physics, not mathematics.
Quoting Mephist
I cannot help but admire your ambition. :smile: You have renamed our map as a plan, something made beforehand to describe what will be made, instead of something created later, to help navigate. So now you have reversed the roles of the map and the territory, suggesting that we actually have the plan and the territory. [ I.e. where the plan is the master/reference, and the territory is a secondary copy. ]
And so, regrettably, I feel compelled to ask the difficult question: where is this "Platonic world", where the plans for the Universe are stored until they are needed? For if the map/plan exists, and this Platonic world is where it exists, then where is this Platonic world? Your surmise seems to rest upon your having an answer to this question, doesn't it? :wink: :chin:
Of course I have! I was only waiting for somebody to ask me to announce the truth to the world :smile: :joke:
The answer of course is based on modern weird physic theories that nobody really understands or is able to verify (string theory, multiverse, 11-dimensional space-time, and similars), so you can use them explain whatever you want!
So, here we go:
- First of all: space-time is "emergent" (https://www.preposterousuniverse.com/quantumspacetime/index.php?title=Emergent_space(time)). This means that what we perceive as space-time could be an "object" (still physical) that has a different metric and even a different number of dimensions (holographic principle: https://en.wikipedia.org/wiki/Holographic_principle). So, there is no way decide which one of the two realities (the ologram the image that it represents) is the really "real" one. It's something similar to what happens with electromagnetic fields in special relativity: the same "thing" can be seen as an electric field by an observer and as a magnetic field by another observer.
- Second: the universe (space-time, elementary particles, forces) that we observe today are the result of an evolution from a much more symmetric (simple) "object", and the forms of what we see today are the result of "choices" of some forms instead of others (https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking). This is similar to what happened with the evolution of life: animals developed into more and more complex forms starting from much simpler inanimate matter.
So, here's the theory: there is exists this very simple "object" that is still there, at the beginning of time (but we don't see it, because from our point of view we can see only contemporary time) and a map, or plan (the "plan" :joke:) made of all possible mathematical objects that can be built. And for some strange reason that nobody knows, from the simple object "emerge" all the forms contained in the map in different quantities: the more symmetrical of them (respecting patterns) are the most common. Obviously, we can see now only part of them, but the global form of the map is some kind of fractal (like giant Mandelbrot set) in which the most common structures are present in all places and at every scale. And of course (to answer the where it exists question) the "emergent" objects are not objects of the physical 3-dimensional space, but what we see as space is only one of them.
P.S. Please don't ask me what's the meaning of the worlds in quotation marks: I have no idea!
The problem with this criticism is that mistakenly equates what is 'out there', in other words what exists in the manifold domain of objects, with the real universe.
But think about this point: that in order to map and understand what is 'out there' - creating the map, that you say is 'mistaken for the territory' - maths itself has proven indispensable. And maths, furthermore, enables science to predict and discover things which could never be known by observation alone.
There are many such cases in modern mathematical physics, such as Dirac's discovery of anti-matter. 'Dirac realised that his relativistic version of the Schrödinger wave equation for electrons predicted the possibility of anti-electrons.' He said, at the time he wrote the paper, 'they must be there', and lo and behold, they were demonstrated before too long.
Eugene Wigner, who has already been cited as author of the celebrated paper in mathematical philosophy, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, won the Nobel Prize 'through the discovery and application of fundamental symmetry principles in sub-atomic physics'. This is what lead him to write that paper (in which, I note, the word 'miracle' occurs 12 times.)
So I think it's not feasible to argue that the relationship between mathematics and nature is merely fortuitous. There seems a deep connection. I think the conceptual problem arises as a consequence of the tendency to regard what is real, as solely what exists in time and space. Hence the question, 'where is this so-called Platonic realm'? And the answer is that it's not in any place; but 'the domain of natural numbers' is nevertheless real, because some numbers are included in it, and others (i.e. the square root of -1) are not.
So there you have an argument for something that is real, but that doesn't exist 'out there anywhere' - which is precisely what makes it difficult to wrap your head around. Our culture is profoundly alienated from such conception (although some mathematical physicists, like Kurt Godel, Roger Penrose and Max Tegmark among others, tend towards platonism.)
There's an interesting SEP article on Platonism in the Philosophy of Mathematics which notes that:
And in fact this anomaly has been noticed by analytic philosophy, which has felt obliged to mount a defense of the 'uncanny efficacy of mathematics' whilst still trying to maintain that it is something that can be understood naturalistically (see this article):
And what our 'our best epistemic theories'? Why, those supported by neo-darwinian materialism! Hence the argument! But they're never going to resolve the dilemma, in my view, on the grounds that number is real, but transcendent in respect to the physical. So maths isn't explained by naturalism, as it transcends naturalism - but as we've basically debarred 'the transcendent' from modern philosophical discourse, then we can't accommodate the idea.
And where does this map exist? Where is the 'place' where this map is stored and retained, ready for later use? The only thing I know of that can store an idea is a conscious mind. Perhaps there is some other container that can also achieve this, but what and where is it, this store?
Maths is indispensable because it's a good and well-crafted tool. It's useful. Also in its predictive power, as you say. For those without satnavs, (real physical) maps are equally indispensable, except for those that are navigating a landscape they know pretty well. Maps are good and useful things, and there's no shame in being the map, not the territory. But a motorway is not a blue line on a paper base. A motorway is a real road, and the blue line on the map is a partial representation of a motorway, drawn for the purpose of aiding navigation. And so it does. The map is a valuable thing. It does not contain actual motorways, but suffers no loss of value as a result.
Quoting Wayfarer
No, not fortuitous. We created maths to be what it is, to do what it does. We made this valuable mapping tool on purpose. It wasn't just luck or random chance.
I do not argue against transcendence, but I wonder where it exists, outside of our minds? And I wonder if my answer is "nowhere"?
An idea is arbitrary data, such as a picture. For this you need a memory to store information, because there are a lot of possible combinations of shapes and colors (or pixels, if you want) that can form a picture, and you don't know which one of them is the one contained in the memory. But often interesting mathematical objects need much less information to be stored that it would seem: the Mandelbrot set is a very good example: just a couple of formulas encode an apparently infinite quantity of shapes. In a sense, the Mandelbrot formula is an excellent image-compression algorithm.
Now the question is: how much information is needed to store all possible interesting mathematical objects? If you want to describe one of them, you need information about which one is it. But to describe all of them, from my point of view this is like describing all possible books that can be written in every possible language: it's just "the combination of all strings of limited length", and this is a complete description.
Now, my idea is: if "interesting" mathematical objects are in some way identifiable by a simple concept, it's conceivable that the description of "all of them" doesn't need information to at all to be described.
I don't believe that the physical universe is "made" of mathematical objects, but that laws of physic in some way favor the development of objects that "resemble" the mathematical structures that we judge "interesting". Why that coincidence? Because the same laws of physics are at the base of the evolution of human mind: in some way the human mind "favours" the recognition of the same structures, because these structures are "favoured" in the whole universe, and our mind is part of it.
Of course there is an enormous quantity of "information" in the physical world around us, and the fact that it's map contains no information doesn't seem to make sense. But the information that we see could be rather related to our very particular "position" inside the universe (something like living in a very narrow subset of the Mandelbrot set).
However this is only a vague idea: you should define how to measure this "information", and even in that case, this doesn't explain where the information about the laws of physics is contained, and probably a thousand of other things that make no sense.
It's not just a tool, or rather, if it is, it's a meta-tool, something used to make tools. Knowing maths, you can make tools you could otherwise not:
Without it, you might have to content yourself with something like this:
Quoting Pattern-chaser
You haven't really advanced an argument for that, though. I think I answered your initial assertion of this point, in terms of the argument that through maths, we can discover many real principles and properties, on the basis of which you can then invent all kinds of devices - like the LHC above. But the things discovered, like natural laws, are plainly not invented by us, and their mathematical qualities are likewise there to be found.
But it's too cheap to just say 'we make it up' - we continually have to validate mathematical ideas, not only against other mathematical ideas, but also against real-world predictions. And so I say that in some fundamental sense, this amounts to a process of discovery, not simply invention, although there are elements of both.
Regarding the transcendent nature of number: number is transcendent in the Kantian sense that it is essential to the operation of reason, whilst not necessarily being explicable by reason. There is no consensus on the nature of number; it is far from easy to understand or explain what it is, precisely, and there are many theories or schools of thought. Yet we use number, often instinctively, to understand and explain many other things.
Yes, we can "discover many real principles and properties" using maths to help, but that does not show maths to be discovered, only the "principles and properties" you refer to.