What a beautiful conversation! I'm wondering for a long time why mathematicians do what they do, and listening to them trying to find this out themselves is fascinating!
I dont mean to be so off topic but does anyone know of a tool to get back into an instagram account? I somehow forgot my login password. I love any tricks you can give me
@Clayton Manuel i really appreciate your reply. I got to the site thru google and I'm trying it out atm. Seems to take quite some time so I will get back to you later with my results.
This is a particularly good interview in Kuhn's 'Is mathematics invented or discovered' series. Chaitin is sincere and seemingly unpremeditated as he grapples with the question, rather than asserting some point of view. I found that this interview opened up the question. And since we can't answer it, this is a good thing.
Man this was fantastic. I get to see Chaitin, the Chaitin of the omega number, have a fascinating conversation about mathematics. Only knew him through books. I fell in love with mathematics around 1993, 26 years later I'm still as fascinated by the beauty of it as I was in the beginning. Thank you so much for uploading this.
@@JM-us3fr Main main interest here is that it seems that proof theory cannot yet establish (even approximate) upper bounds for the proof length for a given unproven statement. So where does a mathematicians' intuition come from and does it work? Is it just trial and error? Does the platonic realm have enough structure to estimate "distances" to other objects?
Always a pleasure to listen to Prof Chaitin. Stephen Wolfram also has an interesting idea about whether maths is invented or discovered (as part of the same series of interviews). I think the prevailing view is somehow converging towards the idea that it is both. For me, fixing the formal system defines the entire possible platonic universe (which resides in the mind of the mathematician). In that sense, the system of axioms is invented, everything else - discovered. I think a loosely formulated foundation can lead to a pretty big platonic universe hence the feeling that we are very often inventing mathematics as we please. All mathematical structures that we encounter when studying maths seem invented but for me any such structure is compatible with the axiom system (meaning they are not arbitrary). What we call mathematics is based on a formal system , inspired by the surrounding world which in my mind explains why it sometimes works well when describing that world, sometimes not. The platonic universe is enormous, probably full of regions filled with valid yet hard to use or relate to concepts. It is the mathematicians intuition that filters those out..
I was jut going to mention Wolfram as well. Chaitins view is very reasonable. There is pure math like calculus. But reality is extremely complex, perhaps derived from fractal like equations which we are unable to find or prove. Bu using computers we can emulate the same behavior. So its not exact an pure math but very good approximations. Pragmatism!
The platonic universe causes difficulties from the get go. Obviously it would be a kind of concept space. Among other things it might be too big. We talk of the class of all sets being too big but it would have to be part of such a concept space as a single unit. It is better of think of mathematics as linguistic conventions with the entailments that follow from definitions - basically a language game The question people are really interested in is mathematical realism. The question may be more pointedly formulated in this way. Does one think that every meaningful statement about the natural numbers(some closed formula in some logical calculus) is either TRUE or FALSE? The mathematician is allowed to define TRUE and FALSE as he pleases as long as TRUE and FALSE are mutually exclusive and collectively exhaustive. A positive commitment means you are a mathematical realist, and a refusal to commit means you are not. I think that will boil down to whether one believes in the natural numbers, whether one believes in the consistency of the Peano Axioms(which we cannot prove by Godel's second Incompleteness theorem) I think that all mathematical claims can be reduced to claims about the natural numbers. So that form of the question might cover everything. As far as calling human mathematics a historical artefact as Wolfram does, Wolfram talks of other axiom systems but even within ZFC, there's an immense deal(an infinite deal) we don't do that we could do. The choice of problems to work on is of course a historical artefact. And the arbitrariness of the axioms we use is not in itself very important when you consider that the claim of mathematics is that such axioms lead to such theorems, never the unconditional truth of the theorems and that is a truth that transcends any specific choice of axioms. So while we choose what problems to work on, the results themselves are absolute in that sense
probably the most honest, or should I say from the heart, absolutely zero on answering the question, but listening to his mind race all over the place trying to put it into words, it felt real.
A fascinating conversation! As a mathematician I easily relate to Chaitin. My take on this is, integers are invented (or defined) then we discover things, like there are building blocks called primes, there's infinitely many of them, etc. It's a bit more complex but that's basically how it works. Now if integers are abstracted from nature (so many fingers, heads of cattle, etc) then mathematical discoveries are in a sense also scientific.
One has to define a set to count the heads of cattle, and this set is arbitrary, then an integer depends on one's decision, it is not a part of Nature.
@@massecl a way to define numbers is to define a number as the class of all sets with a certain cardinality. In that way, natural numbers(and infinities btw) become the equivalence classes of the class of all sets under the equivalence relation of having the same cardinality. Intuitively that would connect the abstract notion of a natural number with Everyday experience, as we associate a set of 5 cows with the number 5 for example.
@@dekippiesip Defining a number by "same cardinality" is circular, just replacing the word. But we can define "same cardinality" by the existence of a bijection. Then the number again depends on the equivalence class, and this class is arbitrary. The number doesn't exist out there on its own, for there is no way out there to decide which class is the case. We may count the heads of cattle, but perhaps the "real world" counts the number of asparagine molecules or of bristles.
@@massecl yes that was where i'm getting at, define 'same cardinality' by bijection(no mentioning of 'numbers' then) and associate every number with an equivalence class under the equivalence relation. This class will then contain every possible set you can think of with that cardinality, including those you mentioned. The only issue could be Russels paradox, since we are sort of appealing to the 'set of all sets'. By calling it the 'class' of all sets I am trying to dodge that, but it may be a cheap attempt.
Integers may be inspired by nature but clearly not abstracted therefrom. Nobody's seen infinite sets or even very large finite numbers. And if math is real(meaning roughly that the Peano axioms are consistent, which we cannot prove courtesy Godel) then clearly this is reality that transcends physical reality - mathematical reality would hold in all possible worlds. But of course it remains an open question whether the Peano Axioms are indeed consistent
First for discovered. Second for cleverly/clumsily recreated to serve our different purposes. Third for recognizing we might never reach the math which inhabits that which is first.
To answer that question with any degree of legitimacy one must first examine the foundations of mathematics for clues. Problem is that the very foundation of mathematics at the moment is suffering from an ongoing potential crisis. So delve deeper into the as yet problematic foundations of mathematics first, and you might just find an answer to your discovery vs invention question, as well. There are many invented notions in mathematics like notations, conventions, definitions, axioms, infinitesimals, limits, etc. There are also countless natural patterns, forces and innate designs in Nature (the physical world) that are inherently mathematical in nature and ultimately independent of our descriptive notations; from our heartbeats to the curvature of spacetime. Also, the fact, for instance, that certain numbers and mathematical conjectures could lie under dubious, convoluted, and/or contrived assumptions/conclusions is a testament to the fact that invented mathematics, albeit erroneous, is possible, at least in theory. Of course, that’s not the kind of mathematics that we’re concerned about here. The kind of mathematics that we should regard as fundamental is one that even an intelligent extraterrestrial culture would independently deem as fundamental and essential for the processing of information and the understanding of the Universe.
At the very last line he finally revealed the terrible truth that he was dancing around for the whole interview, "It's more like a novel than a math paper." Of course, that is only terrible for a mathematician who believed he was doing "god's work."
Nobody said "God's work" and "Mind of God" is a figure of speech for a certain kind of concept space. This isn't religious in any sense you are thinking of
@@jamespower5165 It's been a couple of years, but I doubt that I was bringing anything religious into it, only that he was full of his own self importance, which he then projected onto his work. However, in these two years, I have begun to wonder if most advanced math wasn't "telling a story" rather than finding a mathematical basis for observations. This leads people to wonder about the uncanny accuracy of mathematics, when in reality it is just a story told with a different language.
@blindwillie99 Ok, I have added this to my list of books I need to read. It is first in, first out at position 50, so it may take me a while to get back to you.
If it was discovered, it existed previously. If invented? Mathematics implies a finite length of string is infinitely divisable but observation and experience says otherwise. I'm curious how many mathematical calculations are inherently defective, not exposed to observation or experience to be treated correctly?
Alfred Korzybski: "The map is not the territory." There is a duality that's inherent to math. Mathematics is largely symbolic language. The symbols are invented but the ideas that the symbols represent are discovered. "Fiveness" implies something more than "fourness" with or without a human brain. But without a human brain, fiveness and fourness mean nothing. So there is mathematical reality without us; but it cannot be discovered without the invention of the language of mathematics; and my most important point is that *you cannot finally separate the discovered independent absolutes from the invented language.*
Differential equations is discovered which is sitting there for ever. Take the example of Faraday's law of induction (the third Maxwell's equation) in which Faraday first did the experiment, and then later the third Maxwell's equation was developed with the differential calculus.
Around 9:08 my interpretation is that he's describing a heuristic assumption, where you assume some arbitrary axiom is true and then use what could be systems of equations to solve for an unknown. This could be dangerous, but it also could push past current boundaries, so I agree and I think that would be very useful if a subset of mathematics researchers operated this way, or if all mathematicians were allotted some time for this boundary pushing. The thing that I like is that the assumption can lead to proving itself, but it takes a risky step to get there, like stepping forward into the unknown but then discovering that your step landed on a stepping stone, not the water around it.
This is just the point: the mathematician gets to decide what to take as axiomatic, and see where it leads. Isn’t this just what Riemann and Lobachevski did with Euclid’s parallel postulate? Where’s the problem? On the other hand, Fermat’s Theorem and the four-color mapping conjecture eventually did end up proven as theorems, so who’s to say?
Reality is a local phenomenon defined by our personal light cone (Physics, not wuu juu!). If anything is truly empirical doesn't really matter, so maths is a product of human perception and whatever order we think we can deduct. We cannot know if the rules in math are true in any absolute sense, but we go along the presumption that reality is consistent, though it might not be.
@rwalser You really don't know before later, if at all, that your numbers have any base in reality, but at least you have them to try on your measurements when something new is observed.
Creativity is not a waste of time or life. Mathematics formalizes and makes precise our quantitative reasoning. One can imagine wholly different relationships to quantity in a very general sense, but such reasoning would be - alien to us. Both as a matter of our conditioning sociopsychologically and as a matter of our perception and the physicality that grounds it, our conceptions of number are tied deeply to who we are and how we parse our perceptions. Creativity in the abstract realm can feeling like discovery, because in a sense it is. We develop a set a tools for thinking about the world and thinking about quantitative reasoning itself. Our insistence on consistency requires us to root our concepts in the rules and principles we have developed through history, with the result that as we move into new and more complex numerical relationships or more abstract principles of reasoning or, similarly, more practical and physically focused uses of numbers, we aim to find in each case the approach that does no violence to our understanding of how we think about mathematics. That is, as we extend or ground the system, we find solutions that answer our questions in the ways we generally want to see questions answered. When we make our ideas precise in this way - when we establish new guideposts that advance our quantitative understanding, we need to find mathematically consistent specifications of our new principles. We work to find a way say what needs saying while tying the new ideas to the rules we accept, so in an important sense we work to find "the right answer" to questions people may never have considered before. The sense of discovery is unavoidable, because we do not feel, in mathematics, we have correctly answered our questions until we answer them in ways that comport with our sense of how we reason, how we prove to ourselves we have met all our requirements. Provided we are not completely off track, we find a way to explain and to demonstrate our answers, finally arriving at the formulation that can take on all challenges from within the system and this answer feels necessary precisely because it fits the system. We discover, in a sense, ourselves with each mathematical success. Formalizing our thinking in new mathematical territory both gives us the sense we have found something, so far as we have followed our reasoning while adding to our set of rules and principles in provable, consistent ways, and it results from this creative work of extending our mathematical system. Adding to the toolbox of reasoning is not reducing mathematical work to puzzle solving or gaming in any dismissive sense. It is or can be noble work with profound uses in many fields of human endeavor. We do not require theology to understand this. Leave, also, broader questions of idealism to the side - we do not need to go there. We create mathematics and that creative process involves discovery - of the implications of what we are doing, carrying our reasoning to its conclusions based on what we are and how we make sense of ourselves and the world. There is no waste in that. Chasing your tail to make money to get influence to get power to make more money is a waste of time, a waste of life.
I have some observations and ideas on our mathematical formalism that I'm sure both Gregory Chaitin and Robert Kuhn would be very interested to hear. If I knew how to contact them I would.
I always thought of it as a mixture of both. Pi, for, example, is a constant and a ratio that doesnt change whereas certain maths such as quadratics, are techniques to calculate answers from data.
If math is invented, then we really do have full control over our universe and how we see and understand life. It may also mean that in an alternate universe or timeline, there is something similar to mathematics that is used in place of it, which we would still be able to come up with in this universe.
Intelligence, is one of our Six Eternal Abilities, it stands for 'Logic and Order'. The Perspective-Principle, is one of the basic Creator-Principles, it means, 'All Relations Relationship'. The Perspective-Principle + Intelligence = Mathematic. So, as our Consciousness, gets renewed again and again, Developing-Circuit after Developing-Circuit, the Mathematic also gets Re-Invented, again and again. But our Eternal Life- and Consciousness-Structure, is in it self Mathematic.
this is a modern age version of "god". When somebody claims that something exists in the physical world, we now require some evidence of that something in the physical world.
Richard of Saint Victor wrote: To approve is surely to praise. This one sentence cuts to the quick of thought far more than any mathematical equation can. My question is thus: if math can only work in restricted ways, how can "truth" come out of a deliberately truncated foundation? Seriously...I want to learn......( restricted ways being the set limitations of mathematical constructs....)
Well, if we use the computer to test if an undecidable statement is (statistically) true, then we are essentially probing what is the truth of the statement in the platonic world of mathematics. Just like we probe physics conjectures with experiments. So the platonic world of mathematics stands as the existing background to human mathematical research.
in my mind math is a language that seeks to describe 'relations'. the relations existed well before humans ever came to be, but we can invent more precise 'words' or mathematical constructs to describe those relationships. So in a way math always existed, and we invent it. Depends if you are talking about the subject of reference of the math, or the algorithms/ constructs we invent.
Light is a wave and particle at the same time or maybe we don't have the theoretical instruments to explain it any other way. Maybe we don't yet have the word to describe how did we end up with math. Some talk about math itself as a language and if that may be the case the question can go deeper. Is language itself discovered or invented?
The 'functionality' of mathematics is purely discovered, and is universal. But the 'language' used to describe the discovered mathematical processes is man made (just like we invented language to call a deer a 'deer', or running 'running'). We simply assigned names to the underlying processes that already exist, and then began using them. Functionally, 2 eggs added to a pile of 2 eggs will always equal a pile of 4 eggs. Sure, we can use symbols other than 2, 4, +, =, and eggs, but regardless of what we call them the underlying functionality is always there, and it always holds true. And let's also not forget that mathematics in science is all about ratios - The ratio of time elapsed to say, 1 second. Or the ratio of an amount of mass compared to 1 kg. Or the ratio of the length of some object to 1 cm. Or the ratio of time elapsed to distance traveled (speed, or m/s). All of mathematics describes how one 'something' relates to another 'something'. Even pi, the ratio of the circumference of any circle to it's diameter. Mathematics simply describes how one thing relates to another thing. And it continues to function just the same whether we label and use it or not. Everything that was, is, and will be, is Truth. Everything that never existed, does not exist now, and will not exist in the future, is imaginary (or illusory, or false). Mathematics is a way of relating how one aspect of Truth compares to another aspect of Truth. Either way, I enjoyed the video!:)
The universe doesn't care about our numbers, she'll do what she does one way or the other... Our numbers are our best way to account and predict behavior we observe in an organized and coherent manner. The universe is analog in nature (go beyond planc and what do you get), and numbers are inherently discrete, our best bet at bridging that gap is calculus and that only works in a very limited sense, it's more of a neat trick. Even the smoothest, continuous, well behaved function is discretized with numbers, and assuming it is continuous to match a field is an assumption and more of a show of faith, the second you solve that equations it becomes discrete. I think we'll find in due time that the true reality quantum mechanics is so mysterious and elusive is because our current form of mathematics is incapable of modeling that level of fluidity. We'll get there in time but assuming math in any form is the language of the gods is narcissistic and the best way of stifling any advancement. I think narcissism in general is one of the reasons theoretical physics has really gone no where in 40 years. All advancement has been in experimentally verifying predictions made decades ago. I think in large part because people had the audacity to think they had it all figured it all out then went about trying to prove it, instead of taking what the universe is showing us and going from there.
Just lovely, thanks. Maybe we need to prove time n space exists independently of us in order to invent maths. We can’t create a thought either, so opinion is also impossible. Xx
Thats because spacetime doesnt ! The only explanation for space and time is that they have to be emergent . Its like that episod from Rick and Morty with Universes into universes ...and we cant never see whats outside of our own matter Universe . ;...just by imagination.
as a simple minded kind of guy i am curious as to why there is seemingly a double cross at the center of both our maths and our myths? and should it concern us?
Feinmann pondered this, if a lady were to ask him the question about the fundamental connection between mathematics and reality. "If she's pretty, you discuss it over a few drinks and you can do well with that. I mean, top drawer stuff"
So, a prior is conceptual/not experientially actual ....yet? And a posteriori is empirical physics? But I always though math initiated from this type of brainbody being able experientially observe quantifications of nature's numerical quantities, iow its relative to biophysics, physics, astrophysics etc etc. That its used creatively to invent new combinations that e.g make technologies usable, enhance nature etc. Obviously im not a mathematician 😆. Excellent discussion as usual. Learning heaps from Closer to Truth, thank you... 'a thousand fold' 😁
Science is indeed full of twists and turns.I doubt that we can find inner peace and satisfaction unless we explore the possibility that conflict is unending.
I believe only in the religion of mathematics, and I would not introduce a new "axiom" unless it were PROVEN to be a consequence of ZFC (and if pushed I would include the continuum hypothesis in my fundamental axioms). As some of the most obvious examples why, I would state Skewes' number, or Graham's number; checking finitely many examples for some statement about integers is not sufficient, in the same way that any probability we define for any statement's truth is not sufficient unless that statement is proven true.
If a new "axiom" is proven to be a consequence of an axiomatic system, it is a theorem, not an axiom, by definition. This said, I agree (and had much the same thought) on the "risk of basing something on a non-intuitive axiom that appears to be true" - however the approach is used already: quite a few "proofs" in number theory assume the Riemann Hypothesis, which is far from proven in any mathematical sense, but seems to hold to any empirical test (as did Euler's conjecture on sums of powers... for about 200 years)
Very interesting guy. Seems genuinely worried that he is wasting his life on made up stuff. I was just thinking to myself that that is how religious people feel. And then he says at 4:50 "It's like a religion in a way". I wasted my life on actual religion, so imagine how I feel. Math is certainly more real than any illogical garbage like the Trinity or Hell, or that the hot mess called the Bible is infallibly given by an omniscient Creator. And then at the end the mathematician says "So it's more like a novel than a math paper." Exactly.
Imagine if we were alone on this planet with no previous history, without megalithic structures or ancient writings then we would think we were inventing from nothing.
So how is the Godel anecdote apriori? I agree that it is both invented and discovered. It is for example easy to collapse some maths to other maths. For example some people reduce graph theory to matrices. Further, there is no necessity for maths to be physically useful, however, it had been "incredibly effective", as it goes. Also, as far as astrophysics, if we use two different telescopes to view the same star, it does not count as two discovered stars.😀 Here is a quote from a famous Indian magazine on matrices and graphs. If one can be reduced to another then why do we need two forms of mathematical systems. The reason is a new kind of mathematical system will make certain problems easy to solve. For more example we can solve a ax+by problem with a matrix notation quite easily. So some mathematics is just a transformation from one system to another and not just about "discovery". Btw, I have read Dr Chaitins free book on The Unknowable, AFAIK and it had provided some powerful arguments. I could understand a little. towardsdatascience.com/matrices-are-graphs-c9034f79cfd8 Linear algebra. Graph theory. If you are a data scientist, you have encountered both of these fields in your study or work at some point. They are part of a standard curriculum, frequently used tools in the kit of every engineer. What is rarely taught, however, is that they have a very close and fruitful relationship. Graphs can be used to prove strong structural results about matrices easily and beautifully. To begin our journey, first, we shall take a look at how a matrix can be described with a graph.
@Calum Tatum Yes. Initially maths was useful like for military formations, property management, accounting and calendar management. Then slowly form were abstracted. That sort of Mathematics was usefull. I still know my logarithms and matrices😀😀. But many of the more abstract fields of maths are not of immediate or even future use. The are just discoveries in a platonic land.
@Calum Tatum Yes. Maths is about a sense of beauty among many other things. However, automation theorem proving by machines is quite ugly. But yes, human interest is driven by order, symmetry, elegance and beauty which may not have any practical uses
@Calum Tatum Yes true. For example, if you are used to workflow systems (business flow software. I am into software) in the end there seem to be only two primitives : graphs or tables. The choice of one of these will dictate many aspects of others.
Any proof or work of art etc that can be expressed in our world I think can be expressed as a binary string, right? So I see that as discovery - a kind of algorithmic search for those gems in the vast set of already existing binary strings. More fascinating is the stuff Goedel proved that there are true statements that cannot be proven true. In other words, the proofs of those statements cannot be found in any search of the possible binary strings that we can access in our world. Maybe that is some evidence of a Platonic ideal world beyond our own?
Great video, mathematics are "thoughts in the mind of God" as in laws of nature that explain & determine phenomena and calibrate natural processes. Einstein didn't invent e=mc^2 the energy mass equivalency formula was calibrating the sun's energy billions of years before Einstein discovered it.
"I was in a meeting with creationists and some normal biologists,,, Darwinian biologists" haha what a remarkable way to correct himself. It makes me wonder why Darwinian biologists are considered "normal", when the theory of evolution is yet to provide hard scientific evidence of any kind.
Normal doesn't mean correct. Look up the word normal and I think you will understand. Normal is so misused in conventional speech, as to be nearly meaningless, unless you provide a more complete explanation.
Let's look at the question this way. Could we have evolved, intellectually, scientifically technologically and could the human species have survived without mathematics? There is your answer.
of course mathematics is discovered. its logical implications are like that of a fractal. can you measure the coast of england? concepts only come into existence if they are visited, observed, and described as sound corollaries from what was "known" before. the entire edifice is built on a few axioms anyway. so we discovered a logic fractal. i never understood why this is still a big deal.
If you believe that conciousness is a creation of the material brain then we create math.if you believe that conciousness exists out side of the human brain then we discover math.
Is it even possible to understand what truth is?! If it goes beyond objective reality, that which is amenable to our empirical science, and indeed exists only conceptually, then what are we even considering?
Except for the section on inductively-accepted principles in mathematics, that was not an interesting exchange. Probably the rest of the conversation (from which this clip was extracted) had interesting stuff. And of course Chaitin has produced more interesting stuff in a year than I will produce in a lifetime, but not in this clip.
6:15 Chaitin: "The belief in truth is a fantasy." So much talking, when do we take time to seriously consider the "Human Mindscape ~ Physical Reality divide". I mean if we are the products of Earthly Evolution that we are, there's got to be a solid fundamental reality that's been doing it's thing, independent of our minds, since forever. I listen to "Closer to Truth" and too often it feels like its inextricably trapped within our imaginative mindscapes, and doesn't give actual physical reality more than lip service, before diving right back into our wonderfully entertaining human mindscape. not seeing the forest for the tree, comes to mind
Mathematics has nothing to do with empiricism. Empiricism derives complex physical phenomena from simple ones. Mathematics merely makes up axioms and derives theorems from them using a pre-selected set of rules that can be changed arbitrarily.
On fuck, he might know, but too scared to say. The rhythm changes as he snaps back into status quo . The truth is terrifying. It's why we value fast thought , if you slow down, you will see the assumption
An incipient materialist ? Or at least a sceptic ? I think he is right about the link between religion and platonic idealism ! He is an honest open minded guy trapped in subject specialism ? Has he ever read Hegel , well he should ! Doubting is a better place to dwell a time in than believing over easily !
Its more a question of math being a projection on the world, to help us understand it, as opposed to some 'twoness' existing as a state in the world. That is to say, our concept of moons, our dividing up the world, etc. being a sort of systematic language that we use (And create) to describe and reflect states of the world. But as he describes, pure math is seperate from the world.
I don't know but I think I heard that God works in numbers or through numbers, I'm not sure if that is in the Bible or not but I heard it said so that may be fundamental to everything.
I think mathemaric is neither discovered nor invented, but as human beings and the need for mathematics we just have given a language to express it and make it undrestandable for others based on needs.
Mathematics is a language devised to explain universal truths. I'm no mathematician, but I think that it is commonly said that Newton invented calculus.
I feel like this better question for a mathmetician who is versed in philosophy. Galileo said "mathematic is the alphabet in which God has written the universe"
@Trevor Chase, don’t have a problem with axioms. Mathematical axioms still have intuitive basis for them. For example 2=2 doesn’t require proof as it is intuitively obvious. My problem is with axioms that require massive leaps of faith and from my readings about Godel he seems to have no problems with those.
Too many people even scientists have to find meaning in something. We are getting to the point in science where if the Multiverse theory is actually true if you dig down deep enough the laws of our universe could be the way they are not because of some profound meaning or Reason but because of chance. And a lot of religious people would not understand this but a lot of scientists are not going to be happy if that is the answer.
To me it’s like saying is the alphabet invented, s plus a plus d = a negative emotion. Math is an alphabet created like the alphabet to represent thoughts. Our thoughts.
This man can't really express his deepest feelings. If religion was accepted in academia and not ostracized, this man would probably be openly very religious. But his career and reputation is on the line so he better 'toe the line' or else.
@@joseph_goebbels606 Physics is not the same as mathematics. When creating new mathematics, you are free to choose any axioms you want as long as they are consistent. To say that mathematics is discovered is a bit like saying that when an author writes a book he discovers the story he is writing
I feel the same. I feel that anything abstract requires a physical universe. Abstractions of any kind requiring some sort of substrate to symbolically work with, and maintain it. Every abstraction we use requires physical tools of some sort or another.
The very existence of the idea seems to lend to its reality and discovery. It points to what many have said. Discovery and science are thinking Gods thoughts after him. Perhaps some just hear such statements as a meaningless learned statement of faith repeated to tow the line. When it may have a deeper meaning. That is the problem with catch phrases. People invent them for what may be highly relevant to them. Others just repeat it for no passion at all? And a kind of negative baggage can become culturally attached to it or any statement for that matter. Vanity. All is vanity. I will stop there. Bias and motivation should always be considered. The worst is presumption and the ignorance, self imposed or not, of ones own underlying bias and readiness to attribute motives, poor or otherwise.
X-Files Math is the language of the hostile alien vampires (greed). The capitalist counting corpses can create stark, sterile, space stations floating in emptiness and futuristic bombers. But unlike earthling poets, artists, musicians, mystics, human beings and creators of joy...the capitalist counting corpses that rule US can't create harmony (real intelligence) because vampires (greed) are ignorant (dead). Vampires (greed) who suck the joy out of life have joined the zombies who eat the futures of their children. Zombie Apocalypse is here and happening now.
Truth is one of two possible logic values in one particular binary logic system (there are many more such systems that we just don't use much). You are free to construct logic systems with more than two values or even an infinity of values and people have. We just haven't spent all that much time on exploring whether that leads to interesting results or not.
What a beautiful conversation!
I'm wondering for a long time why mathematicians do what they do, and listening to them trying to find this out themselves is fascinating!
C Oda yes
I dont mean to be so off topic but does anyone know of a tool to get back into an instagram account?
I somehow forgot my login password. I love any tricks you can give me
@Melvin Thaddeus instablaster =)
@Clayton Manuel i really appreciate your reply. I got to the site thru google and I'm trying it out atm.
Seems to take quite some time so I will get back to you later with my results.
@@melvinthaddeus2943 I would presume that without any species that are organisms evolved into having reflective minds,there would be no mathematics.
Nice to listen this man speak. Very humble and honest in his endeavours.
He speaks wonderfully!! And writes even better!
@@LastSonofEther what have u read that you like?
@@Ensource i have read metamaths and currently reading unraveling complexity
I have now decided. This is one of the greatest videos on the internet.
I am a Biology major. This just sounded interesting in the title. What a GREAT interview. Really glad I watched.
This is a particularly good interview in Kuhn's 'Is mathematics invented or discovered' series. Chaitin is sincere and seemingly unpremeditated as he grapples with the question, rather than asserting some point of view. I found that this interview opened up the question. And since we can't answer it, this is a good thing.
Regardless of content - what a moving conversation and Gregory comes across as such a lovely man .
Best channel on CZcams
Facts and London REAL
Man this was fantastic. I get to see Chaitin, the Chaitin of the omega number, have a fascinating conversation about mathematics. Only knew him through books. I fell in love with mathematics around 1993, 26 years later I'm still as fascinated by the beauty of it as I was in the beginning. Thank you so much for uploading this.
Can mathematicians explain their intuitions? Such as *why* they focus on certain research areas and patterns and how much work they put into them?
That would be interesting. I'm not sure why I do mathematics or find certain branches interesting
@@JM-us3fr Main main interest here is that it seems that proof theory cannot yet establish (even approximate) upper bounds for the proof length for a given unproven statement. So where does a mathematicians' intuition come from and does it work? Is it just trial and error? Does the platonic realm have enough structure to estimate "distances" to other objects?
Always a pleasure to listen to Prof Chaitin. Stephen Wolfram also has an interesting idea about whether maths is invented or discovered (as part of the same series of interviews). I think the prevailing view is somehow converging towards the idea that it is both. For me, fixing the formal system defines the entire possible platonic universe (which resides in the mind of the mathematician). In that sense, the system of axioms is invented, everything else - discovered. I think a loosely formulated foundation can lead to a pretty big platonic universe hence the feeling that we are very often inventing mathematics as we please. All mathematical structures that we encounter when studying maths seem invented but for me any such structure is compatible with the axiom system (meaning they are not arbitrary). What we call mathematics is based on a formal system , inspired by the surrounding world which in my mind explains why it sometimes works well when describing that world, sometimes not. The platonic universe is enormous, probably full of regions filled with valid yet hard to use or relate to concepts. It is the mathematicians intuition that filters those out..
I was jut going to mention Wolfram as well. Chaitins view is very reasonable. There is pure math like calculus. But reality is extremely complex, perhaps derived from fractal like equations which we are unable to find or prove. Bu using computers we can emulate the same behavior. So its not exact an pure math but very good approximations. Pragmatism!
The platonic universe causes difficulties from the get go. Obviously it would be a kind of concept space. Among other things it might be too big. We talk of the class of all sets being too big but it would have to be part of such a concept space as a single unit. It is better of think of mathematics as linguistic conventions with the entailments that follow from definitions - basically a language game
The question people are really interested in is mathematical realism. The question may be more pointedly formulated in this way. Does one think that every meaningful statement about the natural numbers(some closed formula in some logical calculus) is either TRUE or FALSE? The mathematician is allowed to define TRUE and FALSE as he pleases as long as TRUE and FALSE are mutually exclusive and collectively exhaustive. A positive commitment means you are a mathematical realist, and a refusal to commit means you are not. I think that will boil down to whether one believes in the natural numbers, whether one believes in the consistency of the Peano Axioms(which we cannot prove by Godel's second Incompleteness theorem) I think that all mathematical claims can be reduced to claims about the natural numbers. So that form of the question might cover everything.
As far as calling human mathematics a historical artefact as Wolfram does, Wolfram talks of other axiom systems but even within ZFC, there's an immense deal(an infinite deal) we don't do that we could do. The choice of problems to work on is of course a historical artefact. And the arbitrariness of the axioms we use is not in itself very important when you consider that the claim of mathematics is that such axioms lead to such theorems, never the unconditional truth of the theorems and that is a truth that transcends any specific choice of axioms. So while we choose what problems to work on, the results themselves are absolute in that sense
This guy is exactly what you think of when you think of a mathematician! (Or if he was a chemist, I could see that too!) Love it!
Somebody give this man an award. Both of them! Bravo!
God Bless them
Which God?
@@ASLUHLUHCE Thee One and Only, YHWH, who is a Triune Deity. Father, Son (Jesus Christ), and Holy Spirit.
You know Him?
fascinating to see an exact person being so emotional and inspired
probably the most honest, or should I say from the heart, absolutely zero on answering the question, but listening to his mind race all over the place trying to put it into words, it felt real.
A fascinating conversation! As a mathematician I easily relate to Chaitin. My take on this is, integers are invented (or defined) then we discover things, like there are building blocks called primes, there's infinitely many of them, etc. It's a bit more complex but that's basically how it works. Now if integers are abstracted from nature (so many fingers, heads of cattle, etc) then mathematical discoveries are in a sense also scientific.
One has to define a set to count the heads of cattle, and this set is arbitrary, then an integer depends on one's decision, it is not a part of Nature.
@@massecl a way to define numbers is to define a number as the class of all sets with a certain cardinality. In that way, natural numbers(and infinities btw) become the equivalence classes of the class of all sets under the equivalence relation of having the same cardinality.
Intuitively that would connect the abstract notion of a natural number with Everyday experience, as we associate a set of 5 cows with the number 5 for example.
@@dekippiesip Defining a number by "same cardinality" is circular, just replacing the word. But we can define "same cardinality" by the existence of a bijection.
Then the number again depends on the equivalence class, and this class is arbitrary. The number doesn't exist out there on its own, for there is no way out there to decide which class is the case. We may count the heads of cattle, but perhaps the "real world" counts the number of asparagine molecules or of bristles.
@@massecl yes that was where i'm getting at, define 'same cardinality' by bijection(no mentioning of 'numbers' then) and associate every number with an equivalence class under the equivalence relation. This class will then contain every possible set you can think of with that cardinality, including those you mentioned.
The only issue could be Russels paradox, since we are sort of appealing to the 'set of all sets'. By calling it the 'class' of all sets I am trying to dodge that, but it may be a cheap attempt.
Integers may be inspired by nature but clearly not abstracted therefrom. Nobody's seen infinite sets or even very large finite numbers. And if math is real(meaning roughly that the Peano axioms are consistent, which we cannot prove courtesy Godel) then clearly this is reality that transcends physical reality - mathematical reality would hold in all possible worlds. But of course it remains an open question whether the Peano Axioms are indeed consistent
First for discovered.
Second for cleverly/clumsily recreated to serve our different purposes.
Third for recognizing we might never reach the math which inhabits that which is first.
EXCELLENT... beautiful explaination... thanks 🙏
To answer that question with any degree of legitimacy one must first examine the foundations of mathematics for clues. Problem is that the very foundation of mathematics at the moment is suffering from an ongoing potential crisis. So delve deeper into the as yet problematic foundations of mathematics first, and you might just find an answer to your discovery vs invention question, as well. There are many invented notions in mathematics like notations, conventions, definitions, axioms, infinitesimals, limits, etc. There are also countless natural patterns, forces and innate designs in Nature (the physical world) that are inherently mathematical in nature and ultimately independent of our descriptive notations; from our heartbeats to the curvature of spacetime. Also, the fact, for instance, that certain numbers and mathematical conjectures could lie under dubious, convoluted, and/or contrived assumptions/conclusions is a testament to the fact that invented mathematics, albeit erroneous, is possible, at least in theory. Of course, that’s not the kind of mathematics that we’re concerned about here. The kind of mathematics that we should regard as fundamental is one that even an intelligent extraterrestrial culture would independently deem as fundamental and essential for the processing of information and the understanding of the Universe.
At the very last line he finally revealed the terrible truth that he was dancing around for the whole interview, "It's more like a novel than a math paper." Of course, that is only terrible for a mathematician who believed he was doing "god's work."
Nobody said "God's work" and "Mind of God" is a figure of speech for a certain kind of concept space. This isn't religious in any sense you are thinking of
@@jamespower5165 It's been a couple of years, but I doubt that I was bringing anything religious into it, only that he was full of his own self importance, which he then projected onto his work. However, in these two years, I have begun to wonder if most advanced math wasn't "telling a story" rather than finding a mathematical basis for observations. This leads people to wonder about the uncanny accuracy of mathematics, when in reality it is just a story told with a different language.
@@jamespower5165These things are not that different.
11:37 ".. because if you believe it then you should disagree with it" ... Godel mark 2? I really like Chaitin.
This might actually be the most fascinating question I've ever seen to ask numerous people.
@blindwillie99 Ok, I have added this to my list of books I need to read. It is first in, first out at position 50, so it may take me a while to get back to you.
What a lovely and honest man. 👍
If it was discovered, it existed previously. If invented? Mathematics implies a finite length of string is infinitely divisable but observation and experience says otherwise. I'm curious how many mathematical calculations are inherently defective, not exposed to observation or experience to be treated correctly?
Alfred Korzybski: "The map is not the territory." There is a duality that's inherent to math. Mathematics is largely symbolic language. The symbols are invented but the ideas that the symbols represent are discovered. "Fiveness" implies something more than "fourness" with or without a human brain. But without a human brain, fiveness and fourness mean nothing. So there is mathematical reality without us; but it cannot be discovered without the invention of the language of mathematics; and my most important point is that *you cannot finally separate the discovered independent absolutes from the invented language.*
Differential equations is discovered which is sitting there for ever. Take the example of Faraday's law of induction (the third Maxwell's equation) in which Faraday first did the experiment, and then later the third Maxwell's equation was developed with the differential calculus.
Looking forward to Berlinski's position on some of these interesting questions... It would be a real treat
4:53 Does anyone know how I can find Godels essays where he talks about this stuff?
Thanks
This is interesting. I think so much of math and physics is beginning to look different than we thought, just 20 years ago.
Around 9:08 my interpretation is that he's describing a heuristic assumption, where you assume some arbitrary axiom is true and then use what could be systems of equations to solve for an unknown. This could be dangerous, but it also could push past current boundaries, so I agree and I think that would be very useful if a subset of mathematics researchers operated this way, or if all mathematicians were allotted some time for this boundary pushing. The thing that I like is that the assumption can lead to proving itself, but it takes a risky step to get there, like stepping forward into the unknown but then discovering that your step landed on a stepping stone, not the water around it.
This is just the point: the mathematician gets to decide what to take as axiomatic, and see where it leads. Isn’t this just what Riemann and Lobachevski did with Euclid’s parallel postulate? Where’s the problem?
On the other hand, Fermat’s Theorem and the four-color mapping conjecture eventually did end up proven as theorems, so who’s to say?
Reality is a local phenomenon defined by our personal light cone (Physics, not wuu juu!). If anything is truly empirical doesn't really matter, so maths is a product of human perception and whatever order we think we can deduct. We cannot know if the rules in math are true in any absolute sense, but we go along the presumption that reality is consistent, though it might not be.
@rwalser You really don't know before later, if at all, that your numbers have any base in reality, but at least you have them to try on your measurements when something new is observed.
I am a physicist and there is no such thing as a “personal light cone”
I lost track in the middle of this conversation. It was quite difficult to understand. Will get back to it much later maybe.
Beautiful and vulnerable
Creativity is not a waste of time or life. Mathematics formalizes and makes precise our quantitative reasoning. One can imagine wholly different relationships to quantity in a very general sense, but such reasoning would be - alien to us. Both as a matter of our conditioning sociopsychologically and as a matter of our perception and the physicality that grounds it, our conceptions of number are tied deeply to who we are and how we parse our perceptions.
Creativity in the abstract realm can feeling like discovery, because in a sense it is. We develop a set a tools for thinking about the world and thinking about quantitative reasoning itself. Our insistence on consistency requires us to root our concepts in the rules and principles we have developed through history, with the result that as we move into new and more complex numerical relationships or more abstract principles of reasoning or, similarly, more practical and physically focused uses of numbers, we aim to find in each case the approach that does no violence to our understanding of how we think about mathematics.
That is, as we extend or ground the system, we find solutions that answer our questions in the ways we generally want to see questions answered. When we make our ideas precise in this way - when we establish new guideposts that advance our quantitative understanding, we need to find mathematically consistent specifications of our new principles. We work to find a way say what needs saying while tying the new ideas to the rules we accept, so in an important sense we work to find "the right answer" to questions people may never have considered before. The sense of discovery is unavoidable, because we do not feel, in mathematics, we have correctly answered our questions until we answer them in ways that comport with our sense of how we reason, how we prove to ourselves we have met all our requirements. Provided we are not completely off track, we find a way to explain and to demonstrate our answers, finally arriving at the formulation that can take on all challenges from within the system and this answer feels necessary precisely because it fits the system. We discover, in a sense, ourselves with each mathematical success.
Formalizing our thinking in new mathematical territory both gives us the sense we have found something, so far as we have followed our reasoning while adding to our set of rules and principles in provable, consistent ways, and it results from this creative work of extending our mathematical system. Adding to the toolbox of reasoning is not reducing mathematical work to puzzle solving or gaming in any dismissive sense. It is or can be noble work with profound uses in many fields of human endeavor. We do not require theology to understand this. Leave, also, broader questions of idealism to the side - we do not need to go there. We create mathematics and that creative process involves discovery - of the implications of what we are doing, carrying our reasoning to its conclusions based on what we are and how we make sense of ourselves and the world.
There is no waste in that. Chasing your tail to make money to get influence to get power to make more money is a waste of time, a waste of life.
I have some observations and ideas on our mathematical formalism that I'm sure both Gregory Chaitin and Robert Kuhn would be very interested to hear. If I knew how to contact them I would.
I always thought of it as a mixture of both. Pi, for, example, is a constant and a ratio that doesnt change whereas certain maths such as quadratics, are techniques to calculate answers from data.
As far as I know, even Pi can be subject to change in non-euclidian space or some other realities
If math is invented, then we really do have full control over our universe and how we see and understand life. It may also mean that in an alternate universe or timeline, there is something similar to mathematics that is used in place of it, which we would still be able to come up with in this universe.
Kurt Gödel was right! Absolutely right!
Intelligence, is one of our Six Eternal Abilities,
it stands for 'Logic and Order'.
The Perspective-Principle, is one of the basic Creator-Principles,
it means, 'All Relations Relationship'.
The Perspective-Principle + Intelligence = Mathematic.
So, as our Consciousness, gets renewed again and again,
Developing-Circuit after Developing-Circuit,
the Mathematic also gets Re-Invented, again and again.
But our Eternal Life- and Consciousness-Structure, is in it self Mathematic.
Awesome!
Thank You!
Peace & Love
I think we invented our interpretation of the mathematics that exists in the physical world
this is a modern age version of "god". When somebody claims that something exists in the physical world, we now require some evidence of that something in the physical world.
This man is so fucking smart and humble!!! Wow!!
Richard of Saint Victor wrote: To approve is surely to praise. This one sentence cuts to the quick of thought far more than any mathematical equation can. My question is thus: if math can only work in restricted ways, how can "truth" come out of a deliberately truncated foundation? Seriously...I want to learn......( restricted ways being the set limitations of mathematical constructs....)
Well, if we use the computer to test if an undecidable statement is (statistically) true, then we are essentially probing what is the truth of the statement in the platonic world of mathematics. Just like we probe physics conjectures with experiments. So the platonic world of mathematics stands as the existing background to human mathematical research.
Asking profound questions is an art form. So is listening.
in my mind math is a language that seeks to describe 'relations'. the relations existed well before humans ever came to be, but we can invent more precise 'words' or mathematical constructs to describe those relationships.
So in a way math always existed, and we invent it. Depends if you are talking about the subject of reference of the math, or the algorithms/ constructs we invent.
Light is a wave and particle at the same time or maybe we don't have the theoretical instruments to explain it any other way. Maybe we don't yet have the word to describe how did we end up with math. Some talk about math itself as a language and if that may be the case the question can go deeper. Is language itself discovered or invented?
The 'functionality' of mathematics is purely discovered, and is universal. But the 'language' used to describe the discovered mathematical processes is man made (just like we invented language to call a deer a 'deer', or running 'running'). We simply assigned names to the underlying processes that already exist, and then began using them. Functionally, 2 eggs added to a pile of 2 eggs will always equal a pile of 4 eggs. Sure, we can use symbols other than 2, 4, +, =, and eggs, but regardless of what we call them the underlying functionality is always there, and it always holds true.
And let's also not forget that mathematics in science is all about ratios - The ratio of time elapsed to say, 1 second. Or the ratio of an amount of mass compared to 1 kg. Or the ratio of the length of some object to 1 cm. Or the ratio of time elapsed to distance traveled (speed, or m/s). All of mathematics describes how one 'something' relates to another 'something'. Even pi, the ratio of the circumference of any circle to it's diameter.
Mathematics simply describes how one thing relates to another thing. And it continues to function just the same whether we label and use it or not.
Everything that was, is, and will be, is Truth. Everything that never existed, does not exist now, and will not exist in the future, is imaginary (or illusory, or false). Mathematics is a way of relating how one aspect of Truth compares to another aspect of Truth.
Either way, I enjoyed the video!:)
I'm an engineer and now I feel my life is wasted
Why? And how?
@@shadowshadow2724 because he never invented (designed) anything, he just discovered it.
@@shadowshadow2724Because Gödel doesn't think much of them.
It's funny. Engineers want to invent things, and mathematicians want to discover them. They feel fooled if it turns out to be the other way around.
7:14 !!!! Yes
The universe doesn't care about our numbers, she'll do what she does one way or the other... Our numbers are our best way to account and predict behavior we observe in an organized and coherent manner. The universe is analog in nature (go beyond planc and what do you get), and numbers are inherently discrete, our best bet at bridging that gap is calculus and that only works in a very limited sense, it's more of a neat trick. Even the smoothest, continuous, well behaved function is discretized with numbers, and assuming it is continuous to match a field is an assumption and more of a show of faith, the second you solve that equations it becomes discrete. I think we'll find in due time that the true reality quantum mechanics is so mysterious and elusive is because our current form of mathematics is incapable of modeling that level of fluidity. We'll get there in time but assuming math in any form is the language of the gods is narcissistic and the best way of stifling any advancement. I think narcissism in general is one of the reasons theoretical physics has really gone no where in 40 years. All advancement has been in experimentally verifying predictions made decades ago. I think in large part because people had the audacity to think they had it all figured it all out then went about trying to prove it, instead of taking what the universe is showing us and going from there.
Just lovely, thanks. Maybe we need to prove time n space exists independently of us in order to invent maths. We can’t create a thought either, so opinion is also impossible. Xx
Thats because spacetime doesnt ! The only explanation for space and time is that they have to be emergent . Its like that episod from Rick and Morty with Universes into universes ...and we cant never see whats outside of our own matter Universe . ;...just by imagination.
Truth is derived from internal and external observations, neither one is an incorrect view because both are necessary.
Internal vs External patterns
Unity vs Differentiation
Categories of Being
as a simple minded kind of guy i am curious as to why there is seemingly a double cross at the center of both our maths and our myths? and should it concern us?
Feinmann pondered this, if a lady were to ask him the question about the fundamental connection between mathematics and reality.
"If she's pretty, you discuss it over a few drinks and you can do well with that. I mean, top drawer stuff"
If math is invented . . . then we humans are god-like in our powers.
Bingo. We are God! HEAVEN is NOW
@@user-js2dr9gv1u Show me the transitive property in nature... in snowdrops. [QED]
So, a prior is conceptual/not experientially actual ....yet? And a posteriori is empirical physics?
But I always though math initiated from this type of brainbody being able experientially observe quantifications of nature's numerical quantities, iow its relative to biophysics, physics, astrophysics etc etc. That its used creatively to invent new combinations that e.g make technologies usable, enhance nature etc. Obviously im not a mathematician 😆. Excellent discussion as usual. Learning heaps from Closer to Truth, thank you... 'a thousand fold' 😁
Science is indeed full of twists and turns.I doubt that we can find inner peace and satisfaction unless we explore the possibility that conflict is unending.
The KINGDOM of HEAVEN is WITHIN
Is it possible that mathematics is a kind of force and not just measurement?
I believe only in the religion of mathematics, and I would not introduce a new "axiom" unless it were PROVEN to be a consequence of ZFC (and if pushed I would include the continuum hypothesis in my fundamental axioms). As some of the most obvious examples why, I would state Skewes' number, or Graham's number; checking finitely many examples for some statement about integers is not sufficient, in the same way that any probability we define for any statement's truth is not sufficient unless that statement is proven true.
If a new "axiom" is proven to be a consequence of an axiomatic system, it is a theorem, not an axiom, by definition.
This said, I agree (and had much the same thought) on the "risk of basing something on a non-intuitive axiom that appears to be true" - however the approach is used already: quite a few "proofs" in number theory assume the Riemann Hypothesis, which is far from proven in any mathematical sense, but seems to hold to any empirical test (as did Euler's conjecture on sums of powers... for about 200 years)
Very interesting guy. Seems genuinely worried that he is wasting his life on made up stuff. I was just thinking to myself that that is how religious people feel. And then he says at 4:50 "It's like a religion in a way".
I wasted my life on actual religion, so imagine how I feel. Math is certainly more real than any illogical garbage like the Trinity or Hell, or that the hot mess called the Bible is infallibly given by an omniscient Creator.
And then at the end the mathematician says "So it's more like a novel than a math paper."
Exactly.
At least math has a firm footing in reality and is useful.
Imagine if we were alone on this planet with no previous history, without megalithic structures or ancient writings then we would think we were inventing from nothing.
So how is the Godel anecdote apriori? I agree that it is both invented and discovered. It is for example easy to collapse some maths to other maths. For example some people reduce graph theory to matrices. Further, there is no necessity for maths to be physically useful, however, it had been "incredibly effective", as it goes. Also, as far as astrophysics, if we use two different telescopes to view the same star, it does not count as two discovered stars.😀
Here is a quote from a famous Indian magazine on matrices and graphs. If one can be reduced to another then why do we need two forms of mathematical systems. The reason is a new kind of mathematical system will make certain problems easy to solve. For more example we can solve a ax+by problem with a matrix notation quite easily. So some mathematics is just a transformation from one system to another and not just about "discovery".
Btw, I have read Dr Chaitins free book on The Unknowable, AFAIK and it had provided some powerful arguments. I could understand a little.
towardsdatascience.com/matrices-are-graphs-c9034f79cfd8
Linear algebra. Graph theory. If you are a data scientist, you have encountered both of these fields in your study or work at some point. They are part of a standard curriculum, frequently used tools in the kit of every engineer.
What is rarely taught, however, is that they have a very close and fruitful relationship. Graphs can be used to prove strong structural results about matrices easily and beautifully.
To begin our journey, first, we shall take a look at how a matrix can be described with a graph.
@Calum Tatum Yes. Initially maths was useful like for military formations, property management, accounting and calendar management. Then slowly form were abstracted. That sort of Mathematics was usefull. I still know my logarithms and matrices😀😀. But many of the more abstract fields of maths are not of immediate or even future use. The are just discoveries in a platonic land.
@Calum Tatum Yes. Maths is about a sense of beauty among many other things. However, automation theorem proving by machines is quite ugly. But yes, human interest is driven by order, symmetry, elegance and beauty which may not have any practical uses
@Calum Tatum Yes true. For example, if you are used to workflow systems (business flow software. I am into software) in the end there seem to be only two primitives : graphs or tables. The choice of one of these will dictate many aspects of others.
Any proof or work of art etc that can be expressed in our world I think can be expressed as a binary string, right? So I see that as discovery - a kind of algorithmic search for those gems in the vast set of already existing binary strings. More fascinating is the stuff Goedel proved that there are true statements that cannot be proven true. In other words, the proofs of those statements cannot be found in any search of the possible binary strings that we can access in our world. Maybe that is some evidence of a Platonic ideal world beyond our own?
But you, O Lord, reign forever; your throne endures to all generations. Lamentations 5:19
Great video, mathematics are "thoughts in the mind of God" as in laws of nature that explain & determine phenomena and calibrate natural processes. Einstein didn't invent e=mc^2 the energy mass equivalency formula was calibrating the sun's energy billions of years before Einstein discovered it.
"I was in a meeting with creationists and some normal biologists,,, Darwinian biologists" haha what a remarkable way to correct himself. It makes me wonder why Darwinian biologists are considered "normal", when the theory of evolution is yet to provide hard scientific evidence of any kind.
Normal doesn't mean correct. Look up the word normal and I think you will understand. Normal is so misused in conventional speech, as to be nearly meaningless, unless you provide a more complete explanation.
Let's look at the question this way. Could we have evolved, intellectually, scientifically technologically and could the human species have survived without mathematics? There is your answer.
of course mathematics is discovered. its logical implications are like that of a fractal. can you measure the coast of england? concepts only come into existence if they are visited, observed, and described as sound corollaries from what was "known" before. the entire edifice is built on a few axioms anyway.
so we discovered a logic fractal.
i never understood why this is still a big deal.
If you believe that conciousness is a creation of the material brain then we create math.if you believe that conciousness exists out side of the human brain then we discover math.
Is it even possible to understand what truth is?! If it goes beyond objective reality, that which is amenable to our empirical science, and indeed exists only conceptually, then what are we even considering?
Except for the section on inductively-accepted principles in mathematics, that was not an interesting exchange. Probably the rest of the conversation (from which this clip was extracted) had interesting stuff. And of course Chaitin has produced more interesting stuff in a year than I will produce in a lifetime, but not in this clip.
Could you explain why it was not interesting?
I disagree.
This man speaks like a s real researcher!
6:15 Chaitin: "The belief in truth is a fantasy." So much talking, when do we take time to seriously consider the "Human Mindscape ~ Physical Reality divide". I mean if we are the products of Earthly Evolution that we are, there's got to be a solid fundamental reality that's been doing it's thing, independent of our minds, since forever.
I listen to "Closer to Truth" and too often it feels like its inextricably trapped within our imaginative mindscapes, and doesn't give actual physical reality more than lip service, before diving right back into our wonderfully entertaining human mindscape.
not seeing the forest for the tree, comes to mind
Just read this quote: "This statement is false". It is basically the same paradox.
one plus one equal two.it is discovered because it was there since the begining
Except there is no "one" or "two" - they are concepts. You can have one cow, or one ball etc., but you can't have a "one".
@@CyberiusT If you are being obtuse, fine, if not, no wonder the world is so F--ked up....
So why was this man never a guest star on _Star Trek_ playing himself?
If empirical / science not real and math measures empirical / science, what does that mean for mathematics?
Mathematics has nothing to do with empiricism. Empiricism derives complex physical phenomena from simple ones. Mathematics merely makes up axioms and derives theorems from them using a pre-selected set of rules that can be changed arbitrarily.
On fuck, he might know, but too scared to say. The rhythm changes as he snaps back into status quo . The truth is terrifying. It's why we value fast thought , if you slow down, you will see the assumption
An incipient materialist ? Or at least a sceptic ? I think he is right about the link between religion and platonic idealism ! He is an honest open minded guy trapped in subject specialism ? Has he ever read Hegel , well he should ! Doubting is a better place to dwell a time in than believing over easily !
The Next question would be" Are natural laws invented or discovered?
Discovered. Although we do invent ways of discovering the discoveries.
My opinion is that if Mars has two moons whether or not there is an observer then mathematics is discovered.
Its more a question of math being a projection on the world, to help us understand it, as opposed to some 'twoness' existing as a state in the world. That is to say, our concept of moons, our dividing up the world, etc. being a sort of systematic language that we use (And create) to describe and reflect states of the world. But as he describes, pure math is seperate from the world.
@@alasdairmacintyre9383 This particular topic is a hard one for me. The question seems to fold back in on itself.
I don't know but I think I heard that God works in numbers or through numbers, I'm not sure if that is in the Bible or not but I heard it said so that may be fundamental to everything.
go f... yourself you theist fuck
I think mathemaric is neither discovered nor invented, but as human beings and the need for mathematics we just have given a language to express it and make it undrestandable for others based on needs.
Mathematics is a language devised to explain universal truths. I'm no mathematician, but I think that it is commonly said that Newton invented calculus.
I feel like this better question for a mathmetician who is versed in philosophy. Galileo said "mathematic is the alphabet in which God has written the universe"
Is pi or not?
Maybe I'm missing his point, but I doubt that most physicists would agree with him that their "discoveries" about the world are actually "inventions".
So Godel had a bias for a priori truths. That is the root of irrational behavior.
@Trevor Chase, don’t have a problem with axioms. Mathematical axioms still have intuitive basis for them. For example 2=2 doesn’t require proof as it is intuitively obvious. My problem is with axioms that require massive leaps of faith and from my readings about Godel he seems to have no problems with those.
Maybe there is only paradox.
Too many people even scientists have to find meaning in something. We are getting to the point in science where if the Multiverse theory is actually true if you dig down deep enough the laws of our universe could be the way they are not because of some profound meaning or Reason but because of chance. And a lot of religious people would not understand this but a lot of scientists are not going to be happy if that is the answer.
To me it’s like saying is the alphabet invented, s plus a plus d = a negative emotion. Math is an alphabet created like the alphabet to represent thoughts. Our thoughts.
Where do our thoughts come from? Are they real? 2+2=4 is empirical as well as factual and real.
This man can't really express his deepest feelings. If religion was accepted in academia and not ostracized, this man would probably be openly very religious. But his career and reputation is on the line so he better 'toe the line' or else.
I have a PhD in mathematics. I feel it is invented. Maybe that's why I stopped doing mathematics?
@@joseph_goebbels606 Physics is not the same as mathematics. When creating new mathematics, you are free to choose any axioms you want as long as they are consistent. To say that mathematics is discovered is a bit like saying that when an author writes a book he discovers the story he is writing
I feel the same. I feel that anything abstract requires a physical universe. Abstractions of any kind requiring some sort of substrate to symbolically work with, and maintain it. Every abstraction we use requires physical tools of some sort or another.
The very existence of the idea seems to lend to its reality and discovery. It points to what many have said. Discovery and science are thinking Gods thoughts after him. Perhaps some just hear such statements as a meaningless learned statement of faith repeated to tow the line. When it may have a deeper meaning. That is the problem with catch phrases. People invent them for what may be highly relevant to them. Others just repeat it for no passion at all? And a kind of negative baggage can become culturally attached to it or any statement for that matter. Vanity. All is vanity. I will stop there. Bias and motivation should always be considered. The worst is presumption and the ignorance, self imposed or not, of ones own underlying bias and readiness to attribute motives, poor or otherwise.
Seems to me that if there is an objective reality, then mathematics must be discovered.
Please give me a formula for jealousy or happiness or love or peace
X-Files
Math is the language of the hostile alien vampires (greed).
The capitalist counting corpses can create stark, sterile, space stations floating in emptiness and futuristic bombers.
But unlike earthling poets, artists, musicians, mystics, human beings and creators of joy...the capitalist counting corpses that rule US can't create harmony (real intelligence) because vampires (greed) are ignorant (dead).
Vampires (greed) who suck the joy out of life have joined the zombies who eat the futures of their children.
Zombie Apocalypse is here and happening now.
Would he accept that truth is a valid construct, and one wonders is math part of an ongoing Creation?
Truth is one of two possible logic values in one particular binary logic system (there are many more such systems that we just don't use much). You are free to construct logic systems with more than two values or even an infinity of values and people have. We just haven't spent all that much time on exploring whether that leads to interesting results or not.
Did the 9th of Beethoven already exist?