I agree Wolfram's statement. I think people study subjects like math and physics and take it on to be some sort of dogmatic religion, failing to realize human biases are also inherent into the knowledge too. Therefore continual improvement is always necessary.
i agree with wolfram as well, you see there is basic maths and then people created frame-works (with some defined rules) to formulate and solve different problems like integrals, matrices etc. Matrices and matrix opererations are also invented with a different set of properties (commutative, associative etc, etc). So linear algebra along with many other frameworks is invented
The boat-airplane analogy is lacking though. We can create games like chess and discover stuff about it which we had not set out to create at the start. Just like we can set the initial axioms of math and discover what follows from those. You might as well say that we discovered chess.
Exactly. This is the main point Frenkel misses. Once we invent the concept of a right triangle, then we can go on to discover properties of our construction, such as the Pythagorean theorem. It's not the theorems that are invented by man, but the mathematical apparatus itself. Once the apparatus is established we can go on investigating its objective properties.
It's not the theorems that are invented by man, but the mathematical apparatus itself. Once man invents the game of chess, anyone can then go investigate properties of the game and discover theorems about it. The same goes for mathematics. Once we develop abstract thinking and labeling systems such as the natural numbers, the consequences will be the same no matter who uses them. As an example, once we invent the concept of a right triangle, then we can use are mathematical apparatus to discover properties of this invention, such as the pythagorean theorem.
The title and the opening statement are two different things. Also, math as a separate entity, and the way a human can understand it, are two different things. One is an absolute unknown, the other is perception. Perception in any organism is simply the way it sees it. Perception, though not invented, is not special.
Firstly, if 2 different persons perceive the same concept (eg, the Pythagorean theorem), that suggests that the perception is real. Secondly, if you say objective mathematical truth is beyond perception and can never be known, then by definition we can never know if objective mathematical truth exists. However, nature, which appears to follow mathematical descriptions, suggests that mathematics (as we understand it) is independent of our perception. Third, the findings of mathematics appear to be non self contradictory.
You're right, we never know. The constellation of neurons indicating consistency is of an apparent unity. That a math equation works is on the order of an evolutionary development. It is like an ability to crawl. But the crawling thing is unaware of any actuality as could be presumed to be truly true. It crawls because it can. We see math because we constellate. The idea that this constitutes knowledge of actual, is requiring the external to become subject to the internal. This cannot be true in a material entirety.. we are shaped by that in which we find shape. We are inseparable, not really real as presumed.
@@finalmattasy *"The idea that this constitutes knowledge of actual, is requiring the external to become subject to the internal."* yes. experience is all we can say that we take from reality. but this brings up a question, which you prompt, here: *"not really real as presumed."* is experience of reality also a part of reality? seems it would have to exist somewhere, and reality seems as likely a place as any. KEvron
I think you're right. I think that "delopmentally" the concept of experience in a humanistic view is contested by experience in a materialistic view. The experience that we view a material unity in science has implications involving communally trustworthy psychological outlooks. Experience is experience, but the religion as it were, in which we assume its boundaries to be contained, is elastic according to the materially connective values of self or being. It's a curious thing to think where our brains may end up in their experience of grabbing hold of (discovering?) that which we see the self as developing (inventing?)
@@finalmattasy *"we assume its boundaries to be contained"* i think a limitation is inescapable; our intersubjective experience of reality and reality, itself, are not the same thing. there's always that degree of separation. now, if the whole of reality, itself, were to become a consciousness.... KEvron
Is a known good historical question, but I think misleading in the sense that may imply mutually incompatibility. A view (mine at least) is that human Math is a language and in that sense an artifact, built on the axioms of choice, that serves beautifully to predict Nature phenomena. Other axioms and laws may produce any number of other Maths, mutually overlaping and consistent, that may be simpler in cases and not in others to predict Nature phenomena. What intelligent life can do is just set the languages (artifacts) to describe and predict, something that is already there (and in that sense discoverable). Or to put it shorter Nature phenomena Math is discoverable, with tools (human math) that are just artifacts, because artifacts, (beautiful and accurate as they are) is all we can process.
If maths is invented, not discovered, then it seems like all numbers should be describable. But the set of all numbers is uncountable. Whereas the set of all describable numbers is countable (since the set of all English sentences is countable).
No one would deny there exists at least one bachelor in the universe. But if I asked whether or not the CONCEPT of a bachelor exists, some people might question it. If concepts exist in some sense, then mathematics exists and is therefore discovered. If concepts don't exist in any reasonable sense, then mathematics doesn't exist and therefore cannot be discovered (nor invented). It really comes down to your preference of what you consider to exist, and beyond that it is just terminology.
*"If concepts exist in some sense, then mathematics exists and is therefore discovered."* this doesn't follow. concepts exist as a product of reason, thus they are created, not discovered. if mathematics is conceptual, then it doesn't follow that it must be discovered rather than created. KEvron
MRTOWELRACK The polydynamics of the movement generates pseudo-autonomy as material property, of the autogenous phenomenon; existing.(...) Simultaneous as my unidimensional variability... unidimensional variability = live-beings
Simon Peters The polydynamics of the movement generates pseudo-autonomy as material property, of the autogenous phenomenon; existing.(...) Simultaneous as my unidimensional variability... unidimensional variability = live-beings
Math consists of Axiomatic Systems we invent. Then within a specific Axiomatic System we discover meaningful statements which we call Theorems if we can prove them. The Pythagorean Theorem is only true in the Axiomatic System of Euclidean Geometry. It is not true in a non-Euclidean Geometry like the surface of a sphere like we live on. Wolfram is correct that there are many possible Axiomatic Systems we could invent and the one's we have invented are essentially an artifact of Sapiens. Indeed, Einstein could not have developed his Relativity Theories with Euclidean Geometry. I discuss this for high school students in a video at my website craighane.com
Colin Dowson - nailed it. Integers are merely a logical continuation of base two relationships at the Planck scale. Universe is binary, which exists independent of our observation... in all reality, what we call "2" is just 1+1. "3"= 1+1+1, etc. Even more fascinating is the structure of primes, within the structure of the Universe. Penrose's twistors give some interesting insights into the origins of mathematical structure. fqxi.org/community/forum/topic/3101
@@JM-us3fr assertions need proof. That's all. No one disputes the effectiveness of mathematics in dealing with the world around us (and beyond). The issue here is a subtle one; is mathematics a 'code' we have invented to describe reality (but is not itself reality), or is maths, as some say, discovered? ''Discovered'' implies that maths is itself the fabric of reality which we uncover.
One good example is integral sign and derivitive signs are invented, It cpuld have been any sign. But the fact that when you draw a tangent line to the bottom of some curve in some xy plane, it will always yeilds line that is parallel to x axis is something that's discovered
This guy has it right. I watched to 4 or 5 such videos, and they all stumbled around, tripping over the "usefulness" of mathematics and the implication in the world of physics, but this guy hits the nail on the head: Mathematical ideas are "out there" waiting to be discovered, plain and simple. The pythagorean theorem is a great example -- it's the universality of mathematics that makes it "real" and otherly. Let's go a little further: all is mathematics. The physical world is, possibly, a manifestation of mathematics -- we are just now discovering the basics, the underlying features, but in the future, just as it is today in physics and most of the other sciences, the language of business, commerce, economics, and law will be exclusively mathematics.
The answer is very simple: Mathematics is both discovered AND invented. The discovered bit is the part that shares a correspondence with the physical world, and the invented bit (like Transfinite Arithmetic) is the part that doesn't. But part of the inherent beauty of mathematics lies in the amazing plasticity, and the unified, cohesive, and internally consistent intersection between the discovered bit and the invented part.
Math is generally ahead of physics. A lot of math work turns out to correspond to physical reality much later then it is theorized. Can we even claim that there is math that never has and WILL NEVER have physical applications?
Dhiraj Budhrani Since i and -1/12 are used in physics there might be more applications than wenn might think. Still, math can describe an infinite number of dimensions and their number is likely finite in physics. Imo.
I think, math is the ivention of how to speak about what we expirience in very general ways. There are no such things as circles or parabolids in real world. These are descriptions of imaginations and therefore inventions.
what?? there are circles in the real world... The number pi.. for example.. do you think is invented?? this number is found in so many places in math, its like intentionally placed there to be found..
@@edro1128 You can also find witches and unicorns everywhere in fairy tales, but this does not prove their reality. Of course fairy tales are a reality.
@@KEvronista Of course is invented but it was not invented by man, that is what I mean... man just discovered and put a name to it and said we did it... Man is arrogant by nature
@@edro1128 if math can be invented by a conscious being, then math can be invented by a conscious being. we are conscious beings, capable of invention. god is redundant. KEvron
I don't believe the blueprint of mathematics exists anywhere. modulo the real obvious physically-realizable things, i believe the "existence" of a new mathematical object is simply the *description* of a logical entailment to what is already established. therefore i would humbly put this question to rest by equating existence with "description" or "visitation" of a new node of logical entailment. note this description can be done by a human or even an automated device. since the combinatorics of logical entailment is infinite (indeed it accelerates with the number of new objects and concepts), we can create a new object by LISTING IT, by simply extending one's arm and picking up a fruit from the combinatorial tree called mathematics. to further digress: we cannot predict nor list the set of new objects that will be eventually known for the same reason we cannot describe the future state of a double pendulum or all the details on the border of a fractal. they obey a local rule, but future states can only be known if each step of the way is visited. mathematics is a kind of growing fractal; new nooks and crannies can only be added to the map if they obey the rules of logic. it is precisely due to its explosive combinatorics that (i) future mathematics is intractable and (ii) one must be an artisan to unlock new logical entailments from what is already known. Of course you can also (try) to use sheer computational power, though it must be guided by a human. the "discovery" process is simply the visitation and description of some statement logically entailed by what is already established. Note branches of this tree may occasionally be refuted and yanked out of the "blueprint". do all rational or prime numbers "exist" ? no!!! are they listable? yes. the moment you compute something or list it, it is added to this fictional blueprint. The universe doesn't care about our crummy listings of logical entailments. In fact the reality of the universe is not even causal (quantum physics). Modus ponens is a macroscopic approximation to the logic of the universe, so this whole discussion could be a figment of human cultural activity. Can you compute the future states of a double pendulum without visiting its trajectory? The answer is no! (non-integrable). So is the case with mathematics. Undecidable, and non-integrable because its wavefront combinatorics is explosive and intractable, the discovery of logic connections between present and past requiring digging, artistry, luck, and computation.
*"The universe doesn't care about our crummy listings of logical entailments."* nice! as i like to say, "the universe doesn't have to carry the one in order to be the universe." KEvron
Math is invented to describe that which is discovered. Thus, correlations and beauty in that which is undiscovered but related to what gas been discovered will naturally be foreshadowed and reflected in the invented math. Imagine watching games of chess being played, and trying to fully understand the entirety of the universe that is chess purely from observing the board and pieces with no other information. You might "discover" that the board has two tones of color, and that these colors form a grid. How do you map the grid? Most you would say to number columns and letter rows, or vice-versa, but that would not be inherent in chess, nor is that your only option. You could opt to use a variation of Spiral Honeycomb Mosaic, according to a finite square grid (addressing goes from large scale to small scale) instead of infinite hex grid. The mere fact that this is possible proves that our ability to describe the world is an invention. Thus, any concepts shown in our description before practical discovery is simply a product of how well our descriptive scheme works in describing the world (assuming the world does not alter itself to fit our belief about the nature of the world).
@@Blackrazor_Daystar But people did invent the symbols "1," "2," "+," "=," etc. People did invent the logical groundwork and the algorithms of addition and subtraction to reflect the truth of the world that if you have one item and then gain one more item that you will then have two, except that is an invented understanding, because where did that that other item come from? Did it not start somewhere else and simply be moved? And there's more! You can literally create systems and algorithms where 1+1 does NOT equal 2. And if we can create those, and they have the same consistency and general structure as the familiar forms of math, then how can you claim that those systems and algorithms are anything but an invention, just simply, we chose the version that is most useful which means, the version that moat accurately reflects our experience of the world? Are mirrors an invention? They also show us a perfect image of the world, and like math, we don't have full control over what a mirror shows, but also like math, we can make a mirror that distorts that image of reality, but we normally don't since we want the most useful mirrors, but still, there are occasions in which a warped mirror is exactly what we want, such as funhouses or the magnified mirror for seeing our faces in greater detail. Mirrors are an invention. The fact that we see reality reflected in the mirror does not change the fact that the mirror itself is an invention.
@@Blackrazor_Daystar the proposition "1+2=3" is an abstraction. abstractions exist as a product of the mind. if you intend to describe anything in reality with that abstraction, then you must construct a set. sets and their descriptions exist as products of the mind. KEvron
So taking a Christian worldview, then God is the same as his son. There is an identity, so that God = his son. So if God is -1/12 and his son is e^πi then -1/12 = e^πi. But e^πi = -1. There's a contradiction since -1 =/ -1/12. However, if God is the Riemann zeta function of his son, then you may be onto something
I don't completely agree,,that Pythagoras theorem is eternal,,because that is a consequence of euclidean axioms of euclidean geometry,,mathematically speaking if you change the definition of "norm" it will change the definition of π,,and also it's value,,,but how you create your axiom ,,that is the due to the creativity of human brain,,I may be wrong,,,but that's what I think
Mathematics had been invented not discovered. Man constricts the numbers to the observable phenomena not the other way around. Once man had no zero, so he inveted it. A while ago imgainary numbers were not existent, so man invented them. And think why numbers fit so well in some theories is because if the square peg does not fit into a round hole the mathematicians will shuffle them in such a way that the peg will finally fit.
Edcademia - Not really. The notation is invented. The natural numbers, however, correspond to real things. If you had no mathematical knowledge at all, you would still have an understanding of the difference between having one of something or two of something.
Mathematics would certainly live in a world of its own, the Platonic Reality, if it was axiom-invariant. Sadly it is not. Presently all that we demand from mathematics is that it should be self-consistent on the basis of a set of axioms we trust to be appropriate for our purposes. Change the axioms and you can "discover" a new mathematics living in another Platonic Reality.
What would be an example of a new or changed axiom, & perhaps what I am asking more specifically stated, is their any fundamental mechanism by which axioms should be adjudicated?
Parallel lines can never converge when extended. That's what Euclidean geometry maintains. Change the axiom and accept that "parallel lines" will converge when extended. You have just created a new geometry, namely spherical geometry.
I think the argument is irrelevant of the initial choice of the axioms and has to do with the 'logic' of any consistent non self contradictory structure i.e. axioms--->theorems--->deductions ... etc... for example any structure (algebraic group , topological transformation , the game of chess etc) , which could contain any actions , procedures and axioms, as long as it is consistent and non contradictory is part of the Mathematical structure and the question is whether the "logic" of it is something people invented or is it a characteristic symptom of 'reality' ? or to put it in another way , would any sentient being come up with the same kind of Mathematics ? ps All of Mathematics according to the Platonic Reality Theory exists in the same "realm" ... Euclidean , Riemannian and all other geometry along with set theory , group theory etc and every theory not yet postulated (regardless of the choice of axioms as long as it fulfills the requirements of being consistent and non self contradictory ... more or less everything described by Goedel as being incomplete is Mathematics )
Therefore a certain form of God or principle exists. We are only making the discovery if we are targeting the best form of math. It is my viewpoint. Could you clarify what is the emphasis of your last sentence: outside of spacetime and logic, what is out there? You meant to say there is nothing out there outside of spacetime and logic (?). Or do you mean to say that mathematics exists beyond spacetime and logic?
The 345 triangle can be used by every civilisation, but it isn't math, it is a knotted rope. Everywhere on Earth, it allows to build a rectangle, but that's only approximate because the surface of the Earth is curved. So the Pythagore theorem is known anywhere, although being false, only for a practical purpose. This example is dubious and pointless.
In what way does Godel argue that it would be impossible for math to have been invented? I feel like mathematics is invented as much as any work of fiction is; it is completely independent of our physical reality and totally in our minds and nothing more. Some of it has its roots in reality since a lot of our mathematics was created to explain reality. Maybe this realm of mathematics is discovery. But the vast majority of it has no relation to reality. And maybe there are as many types of mathematics (i.e. different axiomatic systems) that have no interesting properties as there are possible works of literary fiction (infinitely many?). This makes me feel like the rest of it is really an invention of the mind.
Godel showed that formal systems are incomplete, there are mathematical truths that lie outside any formal systems that try to prove them. We can discover some truths, but Godel showed that there are mathematical truths which we will never discover. So how can we claim to have invented them?
Godel showed if Maths was invented,we would be able to circumvent any obstacles to our inventions but we can't,so it isn't merely invented,it is all a part of Nature unless you think reality is a human invention? Regarding the Integers,yes,they are logical fictions but they represent Ideas that are not invented: We can use Algebra and logic to define "2" so we don't actually need the symbol but it makes life easier if we use symbols but symbols are not the reality they represent!!
I agree Einstein should have stated his view more carefully. Nonetheless, a googleplexadgon probably does not exist anywhere in the universe. But that doesn't mean it's not a valid mathematical concept.
then you truly never learned about the golden ratio ..you cant find it anyware in nature.Its discovered my friend .thinks were there and they are there .See the axioms and apply them on the real world .if a=b and b=c then a=c.First of all how you can change such a think.Second apply this axiom on the real world .The first question that ancient greeks and many people had was WHY ? It happened to start from philosophy but that doesnt mean it wasnt there or its not real .The fact that many people approach it in a philosophic way made many people to think that is invent it.
Math is invented not reality. For there to be a math, there must be an existence or being and that being must be able to have perception(five senses) to able to measure and sense its surrounding for purpose of safety and navigation and survival, mathmatics is language of human survival not language of the universe. Mathematics occur within consciousness, consciousness do not occur within mathmatics, nor there is mathematical equations occurring behind the scene. If you lost your perceptions, then mathematics wouldn't be possible, because when there is no perceptions, there are no dimensions of sound(made possible by air), gravity(feature of distance and time and space), dimension of color(color is made possibly by cells in the eyes), dimension of sentience(made possible by limbic system), dimension of rationality(made possible by neocortex), however these first order materialistic dimensions are different from zone of no-dimension which is zone totally different and we can say its astral we do not yet fully understand. Mathmatics is not real as language is not real, its concept. The universe is not made out of mechanisms which are analysis of natural phenomenons that are labelled within lets say a latin based semantical language which which is adoption from Indian/Arabic/Persian/Babylonian semantical systems for calculation. Universe has no mechanism is direct, conscious and we as humans try to be as close as possibles in our understanding of universe and attune our symbol-based theories and empiricism with universe yet still we have large percent errors and we make another models to correct errors and we make another models to correct models that correct errors within errors lol. Universe was created suddenly and so fast and miracleously and there was no years or time, it was instant. We measure universe with our Gregorian based calendar of 1 rotation with 365 day, however if we measure our carbon dating using ethnic calendars that may have 10000 days in year! or calendar that 12 days per year? what would you think? You totally reconsider the lie of 4.5 billion years old
1. Reason we use the same mathematics, with the same core axioms, is because we're all humans, with similar brains, similar pattern recognition features, etc. This is not evidence that there is only "one" mathematics. Plenty of other systems have been invented. It would be akin to saying our decimal system is "out there", simply because all humans use it, because they have 10 fingers. 2. Where's the evidence to the claim it's timeless and unchanging? We've only experienced a limited amount of time, and even *in* that time, mathematics has changed immensely. I mean until 100 years ago we didn't even know about the existence of, or the importance of imaginary numbers. Why would you say we have the final theory of mathematics, or even a certain field of mathematics, like algebra? 3. Gödel's incompleteness theorem has absolutely nothing to do with this subject. He didn't even give an argument, or explain what he discovered. Worthless.
Pshh... this stupid question... mathematics is a LANGUAGE, therefore it was invented. Now, some of the things expressed by that laguage (mathematics) are indeed "found in nature"... sure... it's like the word "tree", we invented that word to express that "thing"... and in fact there are probably thousands of words for that "thing" that exists indeed. Doesn't mean that language has always existed or that it's this magical thing. The notion that mathematics are some sort of universal constant platonic magical rainbow thing because there is "only one mathematics" is ridiculous, there is only one because that's how the language evolved, that's the convention we (humankind) decided on over centuries, much like now we only use the word "tree" to describe a tree, instead of also using reet or eert or tere, or ñlhdfg or whatever else.
No. We have indeed a language to describe mathematics. But the underlying structures, that we are describing are very much not invented OR comparable to a language. You are confusing our way to classify and represent mathematics as mathematics.
No, that's my point... those "underlying structures" are just that: underlying structures... There's no logical reason to assume that we'd perceive those underlying structures in the same exact way if our brains worked differently, or if we perceived the universe at large in a different fashion other than how we do it. The very discovery of those pre-existing underlying structures depend completely and absolutely on our particular perception, that is to say: the way our brains function; thus they're inherently tied to the languages we've invented to express them. And a third time, just to hammer the point home: Had the language of mathematics not eveloved the way it did, it's entirely possible (and very probable) that we wouldn't understand those "underlying structures" in the same manner we currently do.
Whether humans understand it or not does not make a difference here. Mathematical conclusions are universally true. The fact, that pi is the circumference of a circle with radius 1 is a fact that does not need a human to calculate it for it to be true. You could not change the fact, if you wanted to. Only if you change the meaning of the word circle or pi (which does not make any difference for the maths itself). If you ever took some university classes, you will notice that all of maths is solely based on simple logical rules. Logic and therefore math are entirely independent from any being, that is trying to explore it. The only choice a human has, is what kind of maths he wants to explore, but the conclusions are a result of strict logic and not dependend on the human mathematician.
Nah, I'm sorry... saying "Mathematical conclusions are universally true" is just human hubris. Hell, perfect circles don't even seem to exist in nature, so I fail to see how that could be considered an "universal truth"... they're a great exemple of an invented concept based on pre-existing things. In fact, maybe the reason why pi is such a weird number is precisely because of the fact that it's such an "unnatural" number/concept...
I don't think your interpretation of "universally true" is correct here. "Mathematical conclusions are universally true" refers to the fact that no matter where you are in the universe (or how you choose to describe your mathematics), if you describe an object using the same axioms as we humans use to describe say a circle, then all the properties of a circle can be inferred from it. So the way you choose to describe your mathematics can be considered as a language but the statements you make using that language (i.e. mathematics itself) seem to exist in a "reality of its own". Not that I subscribe to Platonism (or any other view for that matter) but if you think about it, it's actually not that stupid a question to ask :)
None of the interviewees have a deep insight in the topic. They all boast their beliefs and their desires, using dubious and pointless examples. I'd rather have the opinion of a philosopher who worked on the subject, not of mathematicians who have idiosyncratic views, and preach for their parish.
