Philosophy of Numbers - Numberphile 

I love when brady shouts "Seven!". Thanks brady I didnt actually know what a number was thanks for the example

His hairstyle clearly signifies the difference between someone from the Mathematics department and someone from the Philosophy department.

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.

Isn't Platonism the exact opposite of that? Platonism says numbers do exist outside our mind

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.

"You can't see a film about them" Watching a film talking about numbers.

I think he's using a different definition of "exist" than the one we're used to...

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.

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.

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.

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

Numbers, just like other words, are ideas we've created in order to describe and understand the world around us

1:51 excuse me??? That's the complete opposite of platonism!

3:17 I have come to confirm that there are indeed twenty four dinosaurs there. You're welcome.

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.

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.

This was an awesome video! More on formal systems, please!

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.

Always making you think, love this channel.

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.

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.

So your saying numbers go beyond conceptual metaphor and exist outside the empirical. How is that not platonic?

Brady! You've already made a video about the existence of numbers! I remember watching it a few years back.

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.

'How do we know about numbers?' , ultimately comes down to ' How does the brain work?'. Lots of work to do on that.

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.

Haha I am loving this animation style, as well as the talk in this video. Thanks :)

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.

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.

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.

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.

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.

Oh but how could a dinosaur exist if there were no beings around to think the word "dinosaur"??

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

I very much enjoyed the speaker, he's good at getting a point across even when dealing with the non-corporeal.

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.

The mop called and wants it's head back

I dont think Ive ever been this annoyed by a haircut

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.

Numbers are alphabets in a language. Using these alphabets you can create mathematical words and sentences which can be used to communicate an idea.

But do haircuts exists?

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?

Breliant talk! I would like to see more philosophical videos on this channel.

An idea for a follow up video is the philosophy of that hair.

This guy would look a lot better if he completely shaved his head. He'd look less like a hipster

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,

Mark, thank you for not saying "begs" the question!

This question never bothered me....

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!

Thank you this was one of the most intellectually stimulating videos thus far! Love the philosophy and math together! A rare treat!

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."

such a pointy nose.. is that Lemongrab?

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.

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.

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.

Making my day via philosophical video about numbers. :) #bestdayever

I think this guy's a bit wooly. Semantics can make anything "exist".

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...

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 :)

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.

Why is Jean-Baptiste Emanuel Zorg talking about numbers? Oo

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.

I'm studying maths and philosophy at university so I got really excited when I saw this video :D x

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.

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.

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.

I love the animations :D

Sir, please take off that toupé. it's not fooling anyone.

We DO define things into existence. Where else does a family, a concerto, a recipe or a government come from?

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.

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.

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?

Mathematical ontology! An interesting topic motivating many discussions!

The end is an ontological argument. That is amazing.

Maths teachers refused to discuss this at school, and it infuriated me.

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.

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.

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.

+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?

Came here with prospect to know more about numbers but leaving with more chaos about numbers!!

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.

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?

In the example of platonism, where numbers are in our head... Even thinking about a time before thinking allows the process of human thought to be applied to that time. e.g we aren't actually talking about a time before thinking, we're talking about a time before human presence.

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 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.

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.

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.

Numbers seem to be more a "meta" idea. Similarly, if we know about "people" because they are physical, one of the things we know about people is how they feel, their emotions. It's a meta detail about the thing we do know. We also know "how many" physical things there are, in a similar way.

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.

Here's my take: Structure is real. It might even be the only thing we have sufficient justification to call "real". Structure is basically "composed" of relations. Mathematics is a formal system - a language where even the *things* it talks about are defined exclusively in relational terms. What defines the number we write as "2"? Nothing but its relation to all the other numbers and the allowed operations. So mathematics (and formal logic, set theory, category theory, type theory...) is purely relational - and since structure is real, the question of the "unreasonable effectiveness of mathematics in science" is no longer unreasonable. Mathematics is so good at helping us to understand reality because it can be applied to everything sharing the described structure, independent of its physical makeup or anything else in fact. We *invent* novel ways to describe relations which helps us discover (by helping us to conceptualize, recognize and formulate) the occurrence of such structures in the world.

I like the idea that numbers are an abstraction from reality but not fully divorced from it. Using a set of scales to measure an amount of flour, I can see that the flour weighs an amount we call a number, an abstraction from the actual flour itself, but it doesn't mean that the flour's weight is meaningless, socially constructed, or subjective

The physical argument is as good as saying "Penguins cannot fly. I cannot fly. Therefore I am a penguin" which, as much as I'd love, I think I have to dismiss as a real argument.

Defining numbers is like defining a point or a line. You just can not make a meaningful definition of it, but you make the world based on it.

1. Numbers are real, non-physical things 2. We interact with the world through physical interaction ∴ We do not know about numbers That is a point you made, though I believe that is incomplete. We know through understanding. This understand, while it can come through interaction, it can also come through understand. Understanding is inherently non-physical. ∴ we can know about numbers, even if we can't interact with them.

The animations were awesome! Very cute and funny!

o-oh boy rick, this is some mind cracking stuff

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.

Duns Scotus long ago noted that the first property we ascribe to anything is necessarily existence; hence, we ascribe the property of existence to trees, to unicorns, to numbers. This fact illuminates several errors of the exposition here.

There is something uncanny with numbers as I can give you two numbers you've never seen before and you can immediately say which one is greater.

I like this video. It's an important question for 'pure' mathematics because there are different degrees and types of abstraction employed in different areas of math but no unifying concept of what constitutes conceptual validity. For example simple use of the natural numbers, 7+3=10, might be an abstraction, but it's an abstraction towards 'universal application'. Instead of a more concrete statement like, 7 sheep + 3 sheep = 10 sheep, you take a step back and say, this has much broader application than that. 7 somethings + 3 somethings = 10 somethings. That will work for apples, ocean liners, hours and Yen. In fact, you can just leave off the word 'somethings' (which is pretty well understood) and then you have numbers only (pure mathematics). Now personally, I think it's very easy to differentiate between this type of abstraction (whose aim is broad to near-universal application) and something like the square root of a negative number. In both cases we're talking about something conceptually abstracted from the world that physics, for example explores... but the type and degree of abstraction are very different. We can say that their conceptual validity doesn't depend on any relation to the physical world but they are 'removed'--abstracted from the world to vastly different degrees. We need a coherent definition for conceptual validity that applies equally to all mathematical objects.