Philosophy of Numbers - Numberphile
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- เผยแพร่เมื่อ 17 ก.ย. 2015
- We revisit the philosophy department and the question of whether numbers really exist?
Featuring Mark Jago from the University of Nottingham.
More links & stuff in full description below ↓↓↓
Earlier video on numbers' existence: • Do numbers EXIST? - Nu...
Infinity paradoxes: • Infinity Paradoxes - N...
Film and interview by Brady Haran
Edit and animation: Pete McPartlan
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His hairstyle clearly signifies the difference between someone from the Mathematics department and someone from the Philosophy department.
Tom Crawford doesn't agree
Looking at my Institute of Philosophy, I'm not sure either…
Shots fired
@@julianletsche6815
Proof by counter example, lol.
Cool free your mind 😊😊😊
I think this enters the realm of philosophy. According to the philosophical materialism, there are 3 kinds of matter M1, M2 & M3 ordered in levels of abstraction. M1 is what's physical, M2 are ideas and institutions, and M3 are universal truths (scientific and philosophical ones). The paper book is M1, the story of Sherlock Holmes is M2 and numbers are M3. All three levels of Matter (M) really exist, but they support one another like M1>M2>M3. You cannot have M3 without M2 (ideas or Scientific institutions). In the same way, you cannot have M2 without M1 (there are no ideas w/out a physical brain, and there are no institutions without physical people). However, you cannot reduce one Mn into the other. Sherlock Holmes cannot be reduced to the atoms of ink on the paper. This is an actual school of Philosophy in Spain, which main philosopher is Gustavo Bueno.
It does indeed the realm of philosophy. Hence the title and interview with a philosophy expert. :)
+malporveresto That is really interesting to hear about, though obviously what is in each level of abstraction is highly up for debate. How recent is this movement?
So he is quite recent then. I'll have to look him up, thanks :)
Supervenience
+malporveresto Awesome! Simple, logic and exhaustive...Super awesome!!!
Isn't Platonism the exact opposite of that? Platonism says numbers do exist outside our mind
Yea... In the world of forms
+Aeroscience i feel like the video was just edited poorly, because he said that one thing about plato totally out of context... but yeah platonism is that abstract entities are mind-independent.
+Aeroscience He got precisely everything wrong.
Yes, I agree. Platonism is the idea that mathematical objects exist independently of humans and their thoughts. As I have learned, the idea that mathematical objects are human mental contructs is what is called intuitionism, founded by dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer.
Sorry for crypto-comment, but glad I wasn't the only one who thought that. When he was describing what turned out (in his view) to be Platonism I *thought* he was describing constructivism, and was completely Thrown when he said that view was called 'Platonism'. It's like, he's a Platonist but doesn't realise.
Side note: Platonism in math is more an analogous position; it's saying a Platonic'like view is an appropriate way to think about whether numbers are 'real'. It's not necessarily an overall endorsement of Plato's Platonic forms in general.
I love when brady shouts "Seven!". Thanks brady I didnt actually know what a number was thanks for the example
Lol
I live this comment and this too now
First of all, I think the definition presented for Platonism is the exact opposite of what it is supposed to be. Second, the example with dinosaurs would have been much better if it had been replaced with an example before any life existed, as it has been demonstrated that other animals have the ability to count, and therefore dinosaurs probably had this ability too, along with subjective experiences just as we do.
+Eugene Khutoryansky
Animals can't "count" without being introduced to the concept first. No animal knows e.g. the number "seven". They can estimate if one group of items is smaller or larger than another,, or if from a given group (eg young ones) something has been taken away or added to, but they will only "count" if they have been introduced to this concept by humans, as it is done with young humans in school/pre-school.
Regarding your second point, he did establish that it was a hypothetical situation where dinosaurs were unable to do stuff like counting, you could have just used another word like rocks or anything else, so I don't see that as a problem. However, I have to agree with you, Platonism as defined by him is quite different than Platonism as far as I'm aware of. Plato's Forms aren't creations of the human mind.
Yeah, Platonism is closer to what he describes near the end. Numbers are real, non-physical things, such that if they can exist, they do exist.
+Dagobah 359 I'm not really familiar with formalism so I can't say much about how he defined it, but I wouldn't say that Platonism is closer to what he said in the end of the video. If I had to try to formulate a phrase like the one you used to explain Platonism I'd probably go with something like: Numbers exist, are nonphysical concepts, that we access through reasoning. It is hard to explain it in one sentence, if at all possible, and it is definitely a flawed attempt but that's how I would describe it.
Dagobah 359
I hereby define 1 divided by 0 to equal kappa. Now does kappa exist ? (At least it surely ain't a real number)
"You can't see a film about them"
Watching a film talking about numbers.
"Life of Pi". "A Beautiful Mind". The movie about card counters.
I think he's using a different definition of "exist" than the one we're used to...
he is.
+Erik yeah it's an ontological sense of existence. Read Quine's paper: On what there is. very cool stuff.
+Erik Yeah I think we need to define existence before we can go and say if number exist or not.
The model I use place numbers as a construct and not as something physical. Why I separate math from science. I think math is a great tool but just because I can prove something mathematically I can not say it really is that way in the real world. I have to make sure that the analogy of math that I have constructed is not only correct in the mathematical world. But also is the right analogy to use in the physical world. Newtons math was not wrong when he described gravity with it. But he used the wrong analogy (Though he was closer when anyone had been before.)
I think it just gets messy when you start to mix the two things up. Math is great because you can get answer. You can get proves. But Science is great for actually learning about the world. They complement each other greatly.
+Cythil Well put, a very balanced way to look at things imo.
+Cythil do you think mathematics as science tool has limitation to explain things which were categorized as "unknowable" matters rather than "unknown" matters? In my opinion, science has purpose to break the limit of unknown and unknowable.
Maybe I'm being simple, but I think numbers are just abstractions of the real world. We can observe one rock and another rock, and putting them next to each other we get 2 rocks. So I don't think numbers exist other than being abstractions of how the world works.
+ybra Yup, that's what they are. People are over complicating this :P. We know about the world through reasoning and evidence. This guy is ignoring the "reasoning" part.
+ybra How about this, the English language isn't a physical thing... it's a mental construct that we agree upon, a tree isn't a physical thing, it is a word we agree to use to refer to the actual object. But we also have words to refer to more abstract and philosophical things, thoughts, emotions, infinite, contradicitons and so forth. Number is a word aswell... a mental construct, you can have one object, then another object, put them together and you have two objects.
From that logic, where do you get numbers like i, irrationals, infinity?
+goodsire By my logic, numbers is an adjective. Also by Clint's logic right below us.
+goodsire my first point would be that if you ever find yourself talking about someone else's "logic", you should stop and think about what you're doing. with respect to the whole numbers business, you're totally right. even things like pi, that do correspond to things in the real world, can only really be found in abstract geometry
My favorite solution to this whole mess comes from linguistics. A common notion in that field is that language is fundamentally about metaphors. George Lakoff's work focuses a lot on this idea.
When you refer to the word "elephant", you are not referring to a real elephant, but rather an internal representation of what you understand an elephant to be. Elephants are only objective in the sense that it's easy for two people to obtain similar internal models of elephants, but it's still possible to be mistaken about elephants, to have an inaccurate internal model.
When you say there are 24 of something, you are using your internal concept of 24 as a metaphor for some attribute of whatever you're describing. This is really no different than using your internal concept of blueness as a metaphor by calling something blue.
This ties in nicely with the notion of mathematics as a language, and gives a satisfying answer to every major question I've seen on the topic.
Exactly! It's kinda so simple so idk why people are still talking abt this in so many places like Numberphile, so mysteriously, whiteout bringing this up. This seems almost obvious to me, especially once you realize it.
Math is a language, language is complex references to things and relationships beteen them. The symbols we use to represent math are made up by us, just like letters, but the things they describe are relationships between real things like one thing being "2 times" as long as another. "2" itself ig doesn't exist in a vacuum, but the things that do exist relate to each other in this "2" way. And this relationship itself exist relative to all other relationships that exist (or can exist, not sure).
I was also gonna say for example, if there was nothing in the universe but a those two things, then would 3 really exist? But then if there was nothing then no number would exist cuz there is nothing to relate to anything else. Just like if u have no objects then there is no motion, or if there are multiple objects but all the distances remain the same, then saying they are all moving left right or up or down ate all the same as saying they are not moving at all cuz motion is relative. That would mean that if you have nothing then the relationships also don't exist cuz there is nothing to relate to each other. BUT, then I realized that if u have repetition, u could have "two of nothing", which is kinda how set theory defines numbers I think. But ig then the question is, in what context is two of nothing actually "two nothings" (which is when you count the (empty) set itself as an element that can be within other sets, and when is two of nothing just still nothing, in a more physical feeling sense ig.
Oh how's the last 7 years been btw lol. I'd be delighted if u even replied at this point ig lol.
What about theorems in pure math which do not represent anything in the real world? What would they be a metaphor for?
Complete misunderstanding of Platonism, if that's what he's really talking about. Sounds more like Berkeley. Platonic forms aren't just subjective mental creations, they are real things existing independently of individual consciousness. We don't just make them up. Platonism is realist, not really idealist. Aristotle in the metaphysics said that numbers are not forms, but instances of them. Bollocks man.
People here commenting as if they're view is the definitive answer to what numbers are. Really? You're gonna sit and pretend you've just undermined an entire subject of historical and modern western philosophy? There would not be as much discussion between rationalists and empiricists if some bloke on the Internet has the answer.
+AdvancePlays I'm replying to this to make it more visible.
Haha thanks. Also, typo in the first sentence, should be *their :S
+AdvancePlays That's just how humans behave. Is how we discuss. Probably some of those rationalists and empiricists were like that too.
We better not try then, shut down all schools of philosophy. A random guy on the internet said it, there's no point in even trying.
1) people claim to have the answer in these TH-cam comments
2) You claim that they don't have the answer.
3) I claim that you have an incorrect claim
4) Darude Sandstorm.
In order to number things one has to be able both to distinguish members of a set and to define similarity between members of a set. This is not an easy thing to do in the real world. It takes a human mind to define the difference between a dinosaur and the non-dinosaur environment. Even then the more one examines the definition, the less confidence one can have in it. Only subatomic particles are identical entities but the boundaries of subatomic particles are difficult to establish and it is far from clear that they are not simply perturbations of the same field. We divide the world up into things but the boundaries between things are not clear.
100% agree. That is the essence of it all. The qualities of sevenness have no meaning without the context we make for it. Without the borders we draw to circumscribe 7. The seven visible stars - the seven sisters pre-dates us humans but were there 7 stars before? The stars were there without self-consciously being 7. We imbue sevenness to them by arbitrarily circumscribing that part of a sky as a cluster.
+Nick Hill We also have to arbitrarily identify one star from another. Where does one end and another begin? Is a red giant to be counted in the same group as a white dwarf but not a nebula or a galaxy? Where is the boundary of a star that makes it distinct from the next? Once we start to identify separate 'things' to count our definitions must become arbitrarily complex.
herbiepop so perhaps numbers are a branch of set theory, and set theory came into existence when we had brains that started to arrange things into sets. We arrange things into sets before we can count.
Set theory might be the natural phenomena and numbers pop out of that.
In reality, even subatomic particles are not identical entities, as quantum physics says that no two things can occupy the same quantum state. So there is no way to group separate things perfectly into a single quantity, as there will always be differences between them. The only number that would actually "exist" would be the number 1, as there is 1 of everything, of every possible quantum state in the universe.
***** I suspect spacetime is an emergent property of Pauli's exclusion principle. However, this is a tangent to the issue at hand since grouping into sets is subject to arbitrary rules which tend not to account for quantum phenomena. Of course there is endless scope to construct arbitrary rules which limit the members of a set to 1.
1 has existed in our part of the universe since there was a being capable of defining a set where only one of something happened to be a member. Even if that definition were a fleeting thought never recorded.
Definitions of a set include:
How many people capable of killing me are in that cave?
How many children do I have?
It may not even be humans in our part of the universe who were first to mentally construct a frame for a set. And therefore the concept of a count and therefore a concept which magnitude equality could work upon.
Does "Love" exist? Does "War". Certainly people in love do, certainly wars do, but does "love" and "war", in themselves, exist in any meaningful way?
Heck, does "Red" exist? There are red things, and red light, and reactions in the eye and brain upon interaction with light of a certain wavelength that we've arbitrarily deemed "perceiving red light", but does "Red" exist in itself?
It's a tough question, but it's not one that's fundamentally different from the question of whether numbers exist. I think we ought to stop putting numbers on this pedestal and just consider them part of the bigger problem of whether abstract things in general exist. The discussion will be much better focused that way.
If you don't want Numbers on a pedestal, you are totally on the wrong channel.
This is like watching ESPN and saying "what's with all the emphasis on sport?" :)
+Numberphile But this video isn't putting numbers on a pedestal , it is putting existence on a pedestal. All these things, like dinosaurs, red, love etc, including numbers, are just things that brains do to make sense of the world given evolution and enough time.
+chrisofnottingham As it does put existence on a pedestal, it as well puts numbers. This video is more talking about the essence of symbols that we use to build our lives, more concentrated on those symbols that we've relied on equally to our letters. True you can bring in the concept of something like existentialism, but each part has it's right to subject. So do respect the concept of philosophy projected into this situation.
It actually wasn't easy trying to form that response.
It is fair enough to inject a bit of philosophy into into numberfile I guess (tho not especially welcome for me personally). I suppose my point is that you could make much the same video about words.
That I can agree. The deeper it ets, the more we have to stop thinking like humans in the end.
The animations here are really great
0:37 best tree chopping animation. And then it cuts out just before the tree falls on him... lol
Dude tried to figure out the world using physical interactions then he wasn’t there anymore to interact or to figure out lol
Numbers, just like other words, are ideas we've created in order to describe and understand the world around us
3:17 I have come to confirm that there are indeed twenty four dinosaurs there.
You're welcome.
And of course, I continued watching only to see that the dinosaurs are counted for us. My gift to humanity has been destroyed.
+hakkihan tunbak thanks anyway. i, for one, appreciate your effort.
+hakkihan tunbak I believe that you'll find that there are 24 representations of dinosaurs there.
+hakkihan tunbak as science is predicated upon the ability to replicate results, I paused at 3:20 and also got 24.
I paused it at 3:16 and only saw 23 before the last guy snuck in. I was bouta throw hands until I saw your comment and counted again
I believe numbers are a second language we use to describe and analyse the world around us. Everything he discussed would be equally applicable to letters.
+Robert Heath This is, as far as I am aware, the only conceivable answer; so why has it never been covered? The other solutions that have been given are totally flawed
+Robert Heath Letters are definitely constructed by us.
But you can't do any tricks with letters.
You cant put A and B together and create C.
You can with 1, 2 and 3.
There seem to be rules that kinda sorta exist meybe...
But there is no such thing with letters.
***** You can reorder cat to make act... in English.
It doesn't matter what country of origin you are from 2 + 2 will always equal 4.
Reordering letters works differently between countries because they are inventing a system to convey ideas.
Number's already exist and we have (discovered?) them.
Adam Boyd the answer to 2+2 depends on the base (number system) you use. There are thousands of different number systems used around the globe
+abittoogrimforthis Letters are not universal, and within limits of human comprehension and as far as modern knowledge goes numbers are. We have multiple alphabets and different languages, if letters/words were like numbers then people all over the world would have ended up inventing the same language independently of one another. No matter where and when you are numbers are always the same and you can't make them different, they have the same properties, very much unlike letters and language in general which changes constantly.
In a way numbers are less like letters and more like color, or other property of object that can't be separated from it. You can't separate rough or brown from tree bark, but you know that blue and green make yellow, like 2 and 2 make 4.
+Robert Heath In the case of different number system the same symbol just describes different number, nature and properties of number itself remain the same no matter how you denote it.
This was an awesome video! More on formal systems, please!
Haha I am loving this animation style, as well as the talk in this video. Thanks :)
Always making you think, love this channel.
So your saying numbers go beyond conceptual metaphor and exist outside the empirical. How is that not platonic?
+Bad Daddy Vegan I think he defined platonic poorly. To my understanding, it doesn't "exist but only in the mind", it should be "it exists, but on a parallel plane that we can only interact with through thought". With his misunderstood definition, he remained consistent though. Of course.. I'm not an expert in this so I could be wrong.
+noah schaefferkoetter It's not parallel
The Forms are on a higher plane of existence.
Think of them as "more real" than our plane of existence.
If you look at a shadow you can tell it's a person, sure
But if you actually look at the person, it's "more real"
This is what the Forms would be to us; we are the "shadow"
Pseudo Lain
I'm probably not as familar to the forms as you, so I'll take your word for it. IMO it always struck me as more pure... not more real. Slight difference but it would be like saying the Form of a shape is a sphere and all other shapes are imperfect reproductions of that ideal shape. Then other shapes have thier own ideal form and et cetera.
Please let me know how I'm wrong.
noah schaefferkoetter
Nah you've got the general idea, the wording is just finicky.
Forget the numbers, I want to talk about the philosophy of that haircut. It seems to be an actual physical representation of a Calabi-Yau manifold.
1:51 excuse me??? That's the complete opposite of platonism!
u mad bro
+minch333 How is it the opposite? Platonism refers to the existence of abstract objects in another realm. Isn't that what he said in the video?
minch333
Oh, ok. I just clicked the time you provided and right after he says "this is platonism" he actually gives an accurate definition.
Bruno Bessa Oh right!
Great video (even if a good deal of it seems to be either wrong, or poorly explained and/or mischaracterized so it appears to be implied incorrectly). The comments section is spectacular. Scrolling down the page is just one gold mine on another gold mine, and that by itself being rare on youtube is worthy of our collective attention. That the video is making people think and come up with their own variations or ideas, plus come corrections or clarifications means in my eyes at least that it is a smashing success.
I actually already knew all this stuff, and having thought about it quite a bit I still don't really have any answers. I like aspects of this or that doctrine or method or theory, but I have yet to stumble on someone else coming up with something that I think satisfies all my skeptical misgivings on how you categorize abstract universals like numbers or synthetic apriori concepts. Of course I am no where near coming up with anything consistent and rigorous myself to try to answer them. I think that there are at least two sides to the proverbial coin when talking about nature vs nurture in this epistemological way. I have heard it put this way; IS math created or discovered?
One is that math and possibly universal abstracts like numbers are wholly human creations, or at least creations that come only via some sort of intelligence (not necessarily human) which uses them as tools to analyze data and try to come to conclusions regarding it (via modeling reality or counting chicken eggs or what have you). So math and numbers and other universal abstracts are created by intelligent beings, and are used as tools to symbolically map something about reality, whether what is being mapped exists or not.
The other is of course that numbers and math have always existed, as is evidenced by our universe seemingly not coming into being last Thursday so all of the events which happened which lead to our current universe had to follow a logical chain to get us here, and that requires that math is a thing and is more than a mere human tool. We learned math, figured out the truth the system can represent and found out that we can use that language to almost perfectly describe the universe around us in minute detail.
I think both are right, at the same time. I think math was wholly created by humans to explain how logic or any other self-consistent system works, and that the invention works because it happens to correlate to a deep universal truth that we can only sort of describe. MAth, the language and the tool are 100% inventions, but the system they describe is a universal truth that through a lot of trial and error, we were able to use to test our mathematical systems in order to try to see if they work. Lots of things we tried did not work, but the one that best mirrors reality did, and that is the language we call math (and all it entails, like numbers and other universal abstracts).
I have no damn idea how to prove it. I just happen to think it seems accurate, right at this moment.
The way to understand numbers and math is in the context of language.
The question of what math is can be reduced to the question of what language is. In natural languages such as English words usually refer to things or actions in the world, but sometimes words refer to other words as in the case of pronouns. In a mathematical language however, words (i.e. mathematical symbols) can refer to other symbols, as in the case of variables for example, or their meanings can remain open to interpretation.
This "open to interpretation" concept is a bit tricky since it doesn't occur in natural language. So for example the English word "tree", when used normally, refers to physical things that are trees. Trees have certain properties such as having a trunk, being a plant, having a location in space etc. While it's possible to use "tree" other than literally, such as in a poem or by explicitly redefining the word. On the other hand the mathematical symbol (or word) "3" has no fixed meaning. We can use "3" to refer to anything we like, as long as the thing being referred to has the same properties as "3" does in the axiomatic language we are using.
This idea generalizes to any mathematical object, whether it's a quaternion, a group element, a set, or a matrix.
Mathematicians typically seem to find this philosophy offensive because they like to think that mathematical objects have a platonic existence. How dare anyone suggest that the objects of their beautiful mathematical thoughts don't exist? If we want to say that numbers exist, the best we can do is point to a collection of axiomatic systems for numbers and say that they exist as symbols in these systems and that these axiomatic systems, which are formal languages, can be used to talk about things that do exist in the world.
+anon8109 That's a very nice comment, thanks for that.
+anon8109 Mathematical words do not refer to symbols. The definition of a symbol is something that refers to something else. It is the symbol that refers to the concept. The word is also a symbol, which refers to the concept.
A variable is also not open to interpretation. A variable may be a placeholder for an unknown member of a set, but that doesn't mean that it's open to interpretation. It's possible values are totally defined, and your or my opinion as to what it is does not matter.
I think you're right about '3' though. Nothing is inherently '3', not even what we would refer to as three apples, or three people. It is completely up to the speaker to explain the method by which they bring this abstract concept, 3, into the real world, so that others can use the same method and utilize the numerical concept to communicate.
Joe Alias Thanks for the criticism. We could get into a very long detailed discussion about formal logic and semantics.
Logicians say that the meaning of mathematical formulas is based in set-theoretic semantics. But if you try to define set theory you still have to use a formal language, so the situation is a bit circular. If you believe that sets have a platonic existence then this works, but if you take a more skeptical approach, then all there is in math are formal languages that relate to each other, and to the actual world.
anon8109 Yeah the problem of language is a sticky one. Mathematical concepts are supposed to be perfectly pure with no ambiguity as the their properties and rules. But, at some point we must begin talking about them using a language with is by nature ambiguous and up for interpretation.
It seems to me that it's possible for there to be a concept that is so basic and simple that it is effectively possible to communicate that idea even through an imperfect language full of ambiguity. For example, I feel like the idea of a circle - the set of points a fixed distance from a center point - is simple enough that even though language is imprecise, it can easily communicate such a simple idea.
I don't know if this is what is meant by platonic existence. I don't know in what sense the idea of a circle "exists" other than as a concept in someone's mind, but I do feel that it is an idea that is easily communicated.
They way to understand anything is in the context of language. It doesn‘t really matter if numbers exist or not. What matters is that they exist in enough a sense that we can exploit them and make them usefull
'How do we know about numbers?' , ultimately comes down to ' How does the brain work?'. Lots of work to do on that.
That sums it up very well, I think!
4:43 mins: you hit the nail on the head. Abstract numbers/entities are about their potentiality or potential applications. Humans too are representatives of those numbers and their limitless potentialities.
Brady! You've already made a video about the existence of numbers! I remember watching it a few years back.
Numbers are simply a human made symbol for physical objects and phenomena. That's what I think the best explanation is. Anything else is a convoluted way of explaining a pretty easy concept. I believe that we define what we see around us in a way that is repeatable and quantifiable. That's why I think we have math! Everything works together so wonderfully because people worked really hard to make it that way. It represents what goes on around us physically
Oh but how could a dinosaur exist if there were no beings around to think the word "dinosaur"??
+SafetySkull My point exactly, which is also why this is junk philosophy...
+SafetySkull It would still have existed. It just wouldn't be called a dinozaur or anything else.
+SafetySkull +Pivitrix If you somehow took a newborn child to the time of dinosaurs this baby would still be able to interact with the thing we call "dinosaur" smell it, see it, be eaten etc. It isn't about the word we use for it its about being able to prove it exists, which is hard when you can't interact with it.
+Pivitrix But that is Mark's point too, he says the idea that "numbers do not exist before we were there to give them a name" is problematic. The same is also true of the word "dinosaur" but we're discussing numbers here.
Pete McPartlan We haven't even meaningfully defined "exist" how are we supposed to have this conversation?
Did you hear about "Converging Realities - Toward a common philosophy of physics and Mathematics" by Roland Omnès ? He calls his proposal physism. He takes everything you said in your video and abstracts it into one philosophy that is easier to grasp, and doesnt object any of the former ones... a really mind-blowing read in my opinion.
I feel very strongly that numbers are an abstract way to describe physical observations. Like smooth is a way we describe glass, 5 is a way to describe the distinct objects in my pocket. Not only that but we can get a higher level of abstraction by adding or subtracting. Even like combining colors, we can combine numbers. Numbers are attributes of the physical things around us that we have learned to manipulate in so many ways.
The mop called and wants it's head back
+Gryppen The 1990s called and… forget it
+Gryppen its*
I dont think Ive ever been this annoyed by a haircut
I think the "Numbers and Free Will" episode with Edward Frenkel goes well with this episode. Just as Edward was describing how vectors are abstractions represented by a set of numbers from an arbitrarily chosen coordinate system , you could say that numbers are just an abstraction represented in the arbitrary "coordinate system" of the human mind.
Brady, thanks for this video. When you've done other videos on infinity and dividing by zero and irrational numbers I have had thoughts along these lines. I'd be interested to see if Mr. Jago has an opinion on whether or not infinity or rational numbers exist.
My theory:
We humans use base 10 (because we have 10 fingers or whatever), but we can also just use base 2, changes nothing to the question. A number in base 2 is something like …010010111... So basically a sequence of ones and zeros. And what is one and zero? It's information! It's whether something is true or not! What I'm saying is: *Numbers are just a sequence of information*. You can also expand this: Real numbers, equations, operations, it's all information! The world consists of two kinds of stuff: Things/Objects and information about them; a rock weights this much, has this much surface area, etc.
How do we know about information, why should information exist? I mean the Universe exists, and for it to exist, there has to be information, because without information it obviously can't exist. *We know, therefore information has to exist*. Imagine a rock without information..
It's just a theory, what do you think?
+Kametrixom Tikara I disagree, information does not physically exist. Only physical matter and natural processes that govern the interaction of that matter exist. Information about an object, such as a description of that object, whether it be weight, color, etc., are simply summaries of groups of repeatable phenomena as interpreted by human brains. The same goes for labeling particular interesting collections of matter as objects. The brain's capacity for assigning properties to objects is itself a natural process in a complex and reproducible brain system, and this is what actually exists. The existence of object properties independent of human minds is an illusion.
+Thor Is Here Games How do you know that matter and natural processes exist? Humans are only capable of knowing information. Nothing that is not information can be known, nothing that cannot be known can be said to exist.
Sev Mar I think we should be able to define some axioms, things we assume to be true, our existence for example. Without axioms, nothing can be proved because there's no foundation (strongly referring to math here)
Yes that is true, what is key though is to become very conscious of which axioms exactly one bases one's reasoning on. There is nothing more prohibitive to thought than unconscious axioms, such as "matter and physical laws exist because science said so"
+Kametrixom Tikara Wow! Things exist?? Great theory!
But do haircuts exists?
I very much enjoyed the speaker, he's good at getting a point across even when dealing with the non-corporeal.
Thank you this was one of the most intellectually stimulating videos thus far! Love the philosophy and math together! A rare treat!
This guy would look a lot better if he completely shaved his head. He'd look less like a hipster
he's a hipster
An idea for a follow up video is the philosophy of that hair.
So many people in the comments are complaining about this video. Why can't we just appreciate it? Not everything Brady posts will appeal to everyone.
I'm studying maths and philosophy at university so I got really excited when I saw this video :D x
If you look at it from information theory point of view. The universe is made of information. Information can be represented by numbers. So the universe is made of numbers?
For example: you could represent the entire world of warcraft using a single number. This number would include all information of what you/everyone is doing at that moment, and next moment all that happens is that number changes based on some rules (physics).
Same could be done for our universe.
Also, a number is just a way of representing a series of answers to yes or no questions.
Also all of this text you're reading right now was sent to you as a single number... and this video is a number... you and I are also numbers
I don't know what my conclusion is from any of this though... yay numbers!
? determinism isn't required. Also, we know it's not deterministic (unless quantum physics is wrong).
Non-determinism just means I don't know every future/past number based on the present number. But I can still represent the state of everything at the moment as a number.
+Stephen Kamenar All information can be represented as a (possibly infinite) number, but the universe as a FUNCTION must be deterministic. Quantum mechanics tells us we cannot know the future state of quanta, but that might be attributed to our lack of understanding and physical limitations rather than true universal randomness. In theory, the entire lifespan of a quantum particle can be extrapolated from its waveform at a point. There is no conclusive proof that we live in an entirely non-deterministic universe.
bell's inequality shows hidden variables are impossible. It's really random.
the wavefunction of stuff has a deterministic time evolution, yea. But the deterministic state is only showing probabilities, so, that's still random.
Stephen Kamenar Bell's inequality doesn't quite show what you claim. In fact, one interpretation of his theorem is that it requires a fully deterministic universe.
+Stephen Kamenar My understanding of quantum physics is that it requires either for causality or locality to be violated in order for it to make sense.
AKA, the universe either can't be deterministic or can't be localized (as in, there has to be a way for things to interact with things beyond just the speed of light).
Personally, I'll admit that I am not a quantum physicist and do not have a full understanding of the subject, but I do know for a fact that quantum physics is still not well enough understood to state for certain that things are not deterministic.
Also, quantum physics tends to deal with individual particles on a tiny scale. The fact that particles seem to change what they are going to do based on circumstance even with supposedly similar start conditions is what quantum physics is all about. Seemingly random behavior - such as us only being able to chart the probability of an electron being at a certain point around an atom at any given time, rather than knowing the exact location.
Still, random behavior to us does not mean random behavior in actuality. The theory isn't well enough developed to say for sure that things are not deterministic, and to be perfectly honest, the idea that things are "random" to me seems impossible to prove. The only way you could "ever" prove that something happening was happening randomly would be to have the exact same starting conditions on a single particle (quantum physics after all deals with the very tiny) in two separate experiments, and then have the result change.
As far as I know however, this has never been the case, because it is literally impossible to have the same starting conditions for two experiments. Ever. You can have similar start conditions, and ones that are experimentally very similar, but the elapsing of time and the change in forces over quantum-scales seems like it could feasibly explain away what we see as being random.
I mean, we don't even know for sure how gravity is caused. Or the underlying nature of dark energy and matter. Both of these things are likely to be answered by advancements in quantum physics and by physics and mathematics in general, but I think that it's false to claim that determinism is somehow disproved by the current model of quantum physics.
In any case, even if things "are" random, that...doesn't really change anything for us practically speaking. If the universe were deterministic, it would be possible in theory to predict the future by knowing the movements and interactions of every bit of energy out there - but such a prediction is literally impossible within the laws of physics anyway because of simply how much "stuff" is out there to calculate.
We can predict things on a larger scale of course because quantum physics tends to be indistinguishable from ordinary physics as you move away from the smaller scales, but this is obviously more limited in accuracy (which is one reason why weather predictions are often only accurate for days at best).
Also, you mentioned that "I can still represent the state of everything at the moment as a number." Practical concerns aside about actually doing so, quantum physics and ideas such as the uncertainty principle dealing with atomic scales make it quite impossible for us to ever know the state of everything, at least with current understandings. We can know some information, but never all. Some of course is enough in practical terms to do many useful things, but is far from perfect.
Making my day via philosophical video about numbers. :)
#bestdayever
Do numbers exist? Numbers are like a deck of cards that we used for centuries in a practical way to count and add. Then we discovered we could do more with the cards than just shuffle and deal them. We could pile them up and build houses with them. We can do magic tricks. We can use them to play new games. Numbers are like that too. We've had them around for a long time, but we keep discovering new ways to use them, new properties that they have. The properties were always there, but until we looked for the properties, we didn't know they were there,
I love the animations :D
No matter how interesting this topic might be, I missed half of it being distracted by your haircut. Dude, please finish the job and have the deceased cat removed from your skull.
This question never bothered me....
I see many people are trying to reconcile the concept that numbers and Sherlock Holmes are both products of our mind. In the spirit of being clear and concise, I would phrase it like so: Sherlock Holmes, while existing in our brains, is an abstraction; a fractal set of tremendously complex ideas to form a coherent whole. Rational numbers, on the otherhand, are our simplest form of understanding -- a *label* we use to differentiate between quantities of objects in reality. One is an imagined reality. One is a description of reality.
I could be wrong though. Discussion invited.
Breliant talk! I would like to see more philosophical videos on this channel.
such a pointy nose.. is that Lemongrab?
+whoeveriam0iam14222 I find this comment to be UNACCEPTABLE
😂Lemongrab
No that's his cousin...raddishman
+whoeveriam0iam14222 I kept thinking that too. :)
+whoeveriam0iam14222 UNACCEPTABLE!!!, to be honest you only need two things, to resemble him the shape of the head and the nose, it's clear that someone would eventually reproduce him by mere chance.
I personally think that numbers real, but they don't 'exist'. I see them as a unitless representation to be used alongside a unit of measurement. Alone the two are meaningless, but together they gain meaning. So numbers are integral to the structure of the universe, meaning that the universe can't exist without numbers, but numbers can't exist without the universe.
+Azivegu "numbers are integral to the structure of the universe"
Yeah, I'm with you on that. +1 this guy.
***** they exist in combination with a unit. So 5 means nothing really, and neither does cubic meter. But 5 cubic meters means something.
Its not the best example, but just give the idea that numbers have no intrinsic properties. But they do appear. Just think of how e is related to radioactive decay. The number itself means nothing, but there is something special about it.
Not stop hindering my plans to dominate the world!
+Azivegu That's like saying that the universe can't exist without words.
+Zimx02 Only if you define the concept of numbers as a language. I don't think that's what he is doing.
noah schaefferkoetter
And clearly, I'm opposing that because I'm not a big fan of metaphysical hand-waving, and neither should be any scientist or analytic philosopher.
Mathematical ontology! An interesting topic motivating many discussions!
Numbers are alphabets in a language. Using these alphabets you can create mathematical words and sentences which can be used to communicate an idea.
Why is Jean-Baptiste Emanuel Zorg talking about numbers? Oo
I think this guy's a bit wooly. Semantics can make anything "exist".
This sounds like the "if a tree falls in a forest, and no one is around to hear it, does it make a sound" thing.
Is a "sound" something that is heard, or is it compression waves?
Are "numbers" the labels or the things the labels are referencing?
The end is an ontological argument. That is amazing.
Please, stop making these ""Duh... it's simple" comments, people way smarter than you and I spent years thinking of numbers and got to where we are now, a minute at keyboard without any in depth knowledge of subject and study of previous works won't will most likely result in an erroneous opinion. The very nature of philosophy is about questioning things and their nature, there is no useless philosophy, just like there is no useless research because you never know what one can stumble upon while working on a given problem.
this video actually makes me angry. maybe I'm just not smart enough to understand it, but as far as I can tell, this us just junk philosophy, and it really makes me frustrated.
+Rob Mckennie thank you, this is exactly how i felt
Happ MacDonald for some reason I cannot get that link to work on my phone, so I'll try to remember to take a look tomorrow morning on my computer
+Rob Mckennie its frustrating, but its not junk. these theories are not just fun things people came up with, they are forced by various problems that people ran into while studying mathematics etc. Wittgenstein, for instance, started as an engineer and mathematician but came across fundamental problems in number theory and the basis for mathematics which led him to become a philosopher.
+Rob Mckennie In that case you would probably feel that most of philosophy not directly applicable(morality, free will, state and politics etc.) to life is "junk", which is quite a big, if not the biggest part of philosophy. Interestin thingg about philosophy is that as soon as there is a way to gather concrete knowledge on the subject and systemize it it leaves realm of philosophy and becomes science, so most of actual philosophical problems that are left and didn't migrate to the realm of natural sciences are a giant mess of arguments and counter arguments from different points of view without anyone having actual solution so that even people teaching it don't know the right answer. On the positive side, it is one of the few subjects where you can debate with professor about almost anything because of how little in philosophy is agreed upon and proven.
+Tooth PasteEater ethics, useful. political philosophy, useful. free will, not useful, but not junk. talking about if numbers exist, barely anything more than semantics; junk.
We DO define things into existence. Where else does a family, a concerto, a recipe or a government come from?
Please do more philosophy videos! It seems you have a channel for all my interests, but you haven't posted on your philosophy channel in a very long time.
Sir, please take off that toupé. it's not fooling anyone.
+numberphile could you do a video on wether or not it's possible to make a sudoku where you have to get both diagonals as well as all the boxes, rows and columns?
o-oh boy rick, this is some mind cracking stuff
Natural numbers are exactly as out there as trees... Trees only exist in our minds. The things out there are just a bunch of quantum field variations. Abstractions are pretty much *everything* we know about. The really big leap is simply "object". When you say "this is a thing" you have made that jump to abstraction. As soon as you have 'an object' then when you have 'an object' and 'another object' then it's ... two. We've simply labelled them "one", "two", "three", etc.
He says "we can't touch the number 23" but we also can't touch "the colour red" or "high frequency". Numbers are no less physical than any other property. But when air vibrates with high frequency you hear it, when light is red you can see it, and when there are 23 of a thing you can see/hear/feel them. Properties exist and are physical, but can't exist in isolation.
I've seen people really mystify "set theory" definitions, and they were simply adding more layers to this for no benefit. If your axioms reject the natural existence of natural numbers... I reject your axioms. lol
I feel like I'm the only one enjoying these philosophical videos, people who don't enjoy it are straight up ignorants.
Jack Vance has a lovely explanation in his novel "Madouc" -
"A Druidic myth relates how Lucanor, coming upon the other gods as they sat at the banquet table, found them drinking mead in grand style, to the effect that several were drunk, while others remained inexplicably sober; could some be slyly swilling down more than their share?
The disparity led to bickering, and it seemed that a serious quarrel was brewing. Lucanor bade the group to serenity, stating that the controversy no doubt could be settled without recourse either to blows or to bitterness. Then and there Lucanor formulated the concept of numbers and enumeration, which heretofore had not existed. The gods henceforth could tally with precision the number of horns each had consumed and, by this novel method, assure general equity and, further, explain why some were drunk and others not. "The answer, once the new method is mastered, becomes simple!" explained Lucanor. "It is that the drunken gods have taken a greater number of horns than the sober gods, and the mystery is resolved." For this, the invention of mathematics, Lucanor was given great honour."
I think that numbers are just labels we came up with that represents the clusters that we can differentiate when we observe the world. Let's take the number one to begin with: number one is the label for a differentiable entity that we have observed through any of our senses. Having defined how number one emerged, we can deduce how the rest were created. This is the mathematical part which was made up by humans, the part that was invented.
In the other hand, mathematics also describes how these entities, now represented in numbers, interacts with each other or with entities from another cluster. I think that those interactions are inherent on Nature, therefore the mathematical rules that describes those interactions are discovered, because they would be exactly like they are, regardless from the human existence.
Maths teachers refused to discuss this at school, and it infuriated me.
I think Bertrand Russell's Introduction to Mathematical Philosophy is a more clear description (postulate??) of the nature of numbers. It does not rely on Platonism or the 'if it can t exist, it does exist' escape hatch. Russell's paper is available online for free, Google has a link to a site at umass that has the full text in a nice format.
Wish Jago had been this interesting when he taught me first year formal logic in 2004. J/k I passed that module and his band was great.
Came here with prospect to know more about numbers but leaving with more chaos about numbers!!
Mark, thank you for not saying "begs" the question!
The animations were awesome! Very cute and funny!
A number is just a definitive label. Mathematics as a whole is just a language to describe things, whether be real or imaginative. It's similar to any other language in that. Some languages are better at describing some things better htan others.
If I wanted to create a different language, I believe I really could. New words and combinations of sounds, along with rule sets that allow people who learn the language to covey ideas through it. Can I come up with a new math with completely different rules and have it still work? I'm not too sure about that one.
I always find philosphical mathematics really fascinating, it goes deep into logics and I would love to learn a lot more about it ( haven't watched the video yet, but, you know, I am early and have to be hipster and stuff :D )
Have you ever made a video about Maths and Music? There is this interesting "paradox", I think Aristoteles or some other greec came up with it that when you go up (I think) 5 exact octaves (by doubling the frequency) and go the same interval with steps of fifth you don't reach the exact same frequency.
I once heard a maths seminar about it but forgot the details...
Numbers are pictures painted on surfaces that then get loaded with meaning using words that we hear. That is how we interact with them physically. The rules how they interact are meaning loaded pictures too. Numbers are language, like written letters and words. Pretty simple.
He talks about "define something and it exists".
This reminds my to my little pc-games, that i sometimes code my self.
If i define a variable in the code, it starts to exist.
Functions in the game-code can now use it to do some stuff, so it really exists now.
Video titled "Philosophy of Numbers". The first 3 books on the shelf that I noticed: Philosophy, Philosophy 2, Mathematics. Very relevant!
Other notable titles: The Impossible, Quantum Mechanics, Nietzche.
Amazing!
That graphic at the beginning had a guy with a magnifying glass. There's your answer. One herd of dinosaurs looks pretty much like another, but put numbers to those herds by counting (or to the same herd over time) , say 24 and 25, and you have two things that are immediately distinguishable from each other. Numbers are a magnifying glass for the present and a telescope for what's coming. Don't ask about how you know about numbers, ask how they help you know.
Numbers are more like descriptors than "things". We make adjectives into nouns and nouns into adjectives, like "seven" or "white", but just like "white", "seven" doesn't exist by itself, it only describes other things. Similarly, one could say "white" doesn't exist, it's just a human term to define different wavelengths of light. Just like "24" dinosaurs existed in the past, they also had other properties like "big" and "purple" and whatever, but those are just English words we use to communicate their properties as we understand them today. I just don't think there's anything special about numbers that make them any different than any other descriptive word that refers to the properties of a thing.
I feel that "it could exist | it does exist" can also apply to sherlock homes, or any number of things humans have created in their minds. sherlock could easily exist in an alternate universe or even on the other side of our universe.
I think you could argue Sherlock Holmes does exist in the sense that he really is a fictional character. He has physical effects on the world in that people read books about him and may change their actions or thoughts as a result.
I believe that numbers and maths is a way we communicate with the universe. maths and numbers in my opinion are just a languages spoken by the universe and we humans are gradually becoming fluent in this beautiful language.
People aren't very keen with math. They would go and say things like "how do we know numbers if they don't exist in our physical world", and I'd be really curious to know THEIR answer to "How do you understand what I'm saying if the english language doesn't exist in our physical world?"
+Jacopo Barberis Doesn't it exist in our physical world?
If you weren't to learn it from someone, would you find any clues that a rock is actually called a rock?
No.
My view is - abstracts do exist in our physical world, as ideas we hold in our minds. Minds depend on brains, brains are physical.
+Jacopo Barberis The people asking questions like these are mathematicians and philosophers, the people who care the most deeply for the subject. It's hardly academic to dismiss such a significant question.
Tomasz Wota Our minds abstract ideas. So are our minds physical or are our ideas physical?
One thing that came to my mind with the dinosaurs example: "There cannot be 24 dinosaurs because there was nobody there to say in what area." Set enumeration starts with the idea that you somehow have a relation which tells you what are in the set and what are not. And the size of the set is clearly defined by that relation. So there is a notion of relation before there can be notion of enumeration of a set.
It’s been a very long time since I thought about any of this.
I have my own approach to this issue that I have considered more or less reasonable, not knowing if everything thinks exactly I do but I suspect that my way of looking at this can be a sort of ‘formalism’ provided I understand what you mean by possible.
Putting aside for a moment how we know about numbers, and going straight to the question ‘Are numbers real?’ my approach is as follows (and of course it is interesting or I wouldn’t write it). Numbers are a part of reality and are discovered. However, that does not mean that their reality is not bounded to the human mind. Numbers and their being real and discoverable are dependent on the mind but that does not mean that they are ‘created’ or ‘constructed’ by the mind. (how does ‘in’ depend on ‘out’). At the same time, creativity and construction is involved.
Mathematics is associated with the human mental and phenomenological processes of making sense of the world and one component of that is a search for consistency. Numbers are part of the mind’s need for consistency and attending to consistency. In this sense, numbers are real because the mind is real; the reality of numbers are bound with the reality of the mind. This pushes any questions about the reality and nature of numbers to the reality and nature of the mind.
How we “know” about numbers is like asking how we know about consistency. The issue becomes complicated because there are there is a component of ‘intentionality’ (as understood in phenomenology) whereby we make reference to our own ‘moments of attending’ and then make references to those references.
We know about numbers because of our “causal interaction with numbers.” Our interaction with numbers is an interaction with our ‘making sense of the world’ (attending to our attending). It is similar with our use of language to make sense of the world, as abstraction is necessary to language. (I hope that point is clear).
Yes, there are mathematical paradoxes. That is because our symbolic representations of our mathematical intuition that grows out of our mind’s processing and searching for consistency, and the rules associated with them, require or demand a larger framework for that searching and processing. Your talk of “the possibility for something to exist” resonates in this specific way: for a physical object to exist the physical conditions must obtain while for a mathematical object to exist the mind’s processing and searching must allow it. However, unlike an imaginary physical object (e.g. a unicorn) to exist, once the mathematical object is even talked about, then it exists since it is allowed by the mind’s processing and searching for consistency.
Again, what this shows is that any talk about the existence or reality of numbers (and its relationship to knowing about them as well as knowing anything at all) pushes to discussions about the mind and reality itself (or the mind’s relationship with reality). This leads to a discussion about infinity (a discussion that I promise will be short).
Consider ‘infinity’ defined in the following way: Infinity is that which is beyond the mind and conditions the mind. You could think of that as the chemical interactions that make up the brain, or the physical conditions that allow us to exist at all, or the integrity of the entire universe. All of those things are things that we do not fully understand, which is precisely the point. The mind exists on the conditions of a set of factors that is beyond the mind’s attending. Or, the conditions allow the mind’s attending to exist and be as they are. Again, infinity is a word to allude to that which conditions the things that we perceive, attend to, and contemplate. It is that which conditions the mediation or relationship between the mind and that we perceive, attend to, and contemplate. At the same time, using the word ‘infinity’ is a way to attend to that which is beyond our attending, and it is that which constitutes the conditions that allows the mind to be what it is. (It is the noumena).
Because the mind is real, it is conditioned, and because it is conditioned, then mathematics has form that we are able to discover, despite the fact that its existence depends on the mind.
Occam's razor suggests that the simplest answer is probably the correct one. In my opinion, Mathematics is simply a language humans use to describe objective reality whereas English, French, Spanish, etc. are languages that humans use to describe subjective reality. We use common language to communicate ideas about what we subjectively observe and we use Mathematics to communicate ideas we objectively prove to be true.
Something I'd like to suggest is that numbers might exist as adjectives, not as nouns. If you say "The red balloon", red is describing the balloon, and if you say "There are three geese", three is describing the geese. "Red" isn't an object. We may have a red apple, or red paint, but we don't ever just have red. So numbers could be the same way. We can have three pennies and three lampshades, but we don't ever just have three. And when we think about equations involving a variable, numbers are just describing whatever "x" happens to be. So maybe by doing math, we're really just calculating the modifiers on tangible quantities. And because numbers are only adjectives, they don't have to actually exist themselves.
This isn't meant to be an end-all-be-all solution, but I do think it's something worth considering :)
pretttyy cool how this is still basically aristotle frowning thousands of years into the future
I would say numbers are like colors. They are discibing real objects and can't exist on their own.
Numbers arise from geometry. From physical space of infinity and infinite numbers that become discrete though form.
So if there is a set that does not exist, then to trade one's items away into another kind of form becomes that set, thus, business is a remainder set of the trade that we wish to do?
Oh dear! I think this one needed more preparation. Was it a problem with editing that led to the mixed up and contradictory description of Platonism, or was he just confused?
I thought you had a separate channel for philosophy and religion? I come here for science.
I think numbers are symbolic descriptors and logic is a connector we use to give meaning to numbers. I'd say it's more important to talk about where logic comes from than where numbers come from.