# What are Numbers? Philosophy of Mathematics

Hello and welcome to Elucidations, a philosophy

podcast recorded at the University of Chicago, I’m Matt Teichman & I’m Jamie Edwards.

With us today is Daniel Sutherland, Professor of Philosophy at the

University of Illinois at Chicago. And he’s here to talk to us about

the philosophy of mathematics. Daniel Sutherland, welcome.

Thank you. Thanks for having me. One of the perennial questions in

the philosophy of mathematics is: “What is a number?”

Maybe we can just start there. Well, that’s a very good question.

Numbers appear to be abstract objects. And by ‘abstract object’ philosophers

mean objects that don’t seem to occupy a spatial or a temporal location; and

also a kind of thing that doesn’t enter in causal relations with other things. And

in fact, numbers are one of the prime candidates for abstract objects, should

there be any. Now, various people have been quite puzzled about the

role of abstract objects in general: Whether there are any–there are people

who have been opposed to admitting that there are abstract objects and have tried to

explain away cases of apparent abstract objects, and that’s the case in the

philosophy of mathematics as well. One of the troubling things about allowing

for abstract objects is that it’s hard to see how they fit into our conception of

the world as we understand it today, that is to say, through science, where

we try to explain things through physical laws and causal interactions. And that

has given a lot of people motivation to resist the idea of abstract objects

altogether. Another more pressing problem within philosophy is to explain how it

is that we could have knowledge of abstract objects and their properties

if they really are abstract. If they don’t occupy a spatial & temporal location,

if they don’t enter in causal relations, then how do we have knowledge of them? This issue has actually been

tremendously formative to philosophy itself because it looks like,

typically, the way you know something is to stand in some sort of causal relation

to it. So the paradigm case here would be perceiving the things in the

room that you’re in right now. Light strikes an object, it reflects

off of it, it impinges on your eye, it causes neurons to fire. That’s a more

detailed scientific picture that we have now than was available at an earlier

point of time, but still, the idea is that you enter in some sort of causal

relation to an object, and that’s what justifies your knowledge that, say, for

example, there’s a cup on the table right now. Now if we’re talking about an abstract

object, it is, by definition, something that doesn’t enter into causal relations.

So, how can you have knowledge of it? Plato and other Rationalists thought that

you have some sort of perception of things that are abstract, and described in

perceptual metaphors our knowledge of the Forms — which you may’ve heard

about — and also numbers. These are ideal, abstract objects that we don’t enter

into causal relations with, or at least no direct causal relations with. So

that raises a puzzle: Are you attributing to human beings a distinct mode of

perception? And if so how do you explain it? And that has been a big problem

for those who believe that you have knowledge of abstract objects, is

giving an account of the epistemology. That is to say, the theory of knowledge, giving a

theory of knowledge of these abstract objects. How is it that you can know things about

them? This problem has perplexed people so much that some have looked for

alternative views about what numbers are. They have tried to locate numbers in the

mind or perhaps in the physical world. One way to do this, and this is a

simplified way of thinking about it. You might think that numbers are in the

world in the sense that the number two is really summarizing or somehow an indirect way of thinking about two

things, two things that you experience in the world. That’s to give numbers some

sort of a concrete realization and then what you know is something in the world.

There are much more sophisticated versions of making numbers a part of

the world that I won’t get into. Another idea is to say that numbers are

constructed by us. They’re in some sense just in our minds. And there are

different approaches in the philosophy mathematics corresponding to the

different attitudes about what a number is and theories about how we know

numbers. So, if I step back a bit to what I was saying at the beginning. Some philosophers think that numbers are

abstract objects, and those philosophers have a way of thinking about numbers

that seems very natural. We think of THE number 3, we don’t think of 3

things here, 3 things there. We think that there is THE number 3. And

it also happens to be a way that most mathematicians seem to think about

numbers, that they are objects. And it also fits very neatly with our theories

of truth. I won’t go into the details there. But there’s a way in which, if you

just take numbers as abstract objects, it seems to follow from the way we talk

about numbers that they’re abstract objects, it fits with a very natural conception of

them, and it fits with our theory of truth. But if you take that option, then you

have to explain how we KNOW these abstract objects. So, your work

then is gonna be cut out in the epistemological side. Those who claim

that numbers are really manifested in the world have an easier time

explaining how you know numbers. Oh, you know numbers by seeing

collections of 2 and then maybe you abstract out an idea of 2, and

when you’re talking about 2 + 3=5, that’s shorthand for groups of

2 put together with groups of 3 give you groups of 5.

Something like that. But it makes the epistemology more

tractable. The problem is is giving an account of numbers that really fits the

way we think about numbers, the way we talk about numbers, the way we

theorize about numbers in a way that gives us a comprehensive theory of what

numbers are. And that requires a lot of work. Plus you have to, even in the

epistemology, explain how you get from seeing 2 things to the number 2. And

that turns out not to be as easy as you might think. It looks like it’s a quick

step, but there’s a big difference between having a collection of 2 and

talking about the number 2, and some complicated relations between them.

Would you say just a little more about this? This seems like we would have the

same issue with colors for instance, abstracting the color blue away from

blue things. It seems like this would be a similar step we would have to make.

But people don’t seem particularly worried about making that step with colors, why would we be worried about

making that step with numbers? That’s a very good question. So an answer

to this actually pulls us into questions about the status of what it is that you abstract

out and what you’re talking about then. So some of the same issues may arise

here. There’s two issues here, one is abstraction and then there

is being an abstract object. So you can abstract — there’s the

abstraction process that might get you the idea of the number 2, an abstraction

that gets you the idea of the color blue. Then there’s a further question,

what those correspond to. Is the one representing an abstract object, the number

two? And then, what about the color blue? What is the status of the color blue

whose representation you’ve abstracted from particular instances of blue?

Is that something that exists independently of the things that it’s

instantiated in? So what we’re seeing here is that the issues in philosophy

of mathematics are closely tied to perennial issues in philosophy about

how to think about universals and how to think about the process of abstraction and what the content of our thoughts are

after we’ve engaged in a process of abstraction. So some of the same issues are

on the table. I think in the case of mathematics it’s particularly problematic

because the idea is that we seem to be talking about an abstract object, and

that gives the problem more bite than it might otherwise have. But there are

similar issues in both cases that would need to be addressed. And I’m glad

you raised this because the problem of universals — this classic problem in

philosophy — you have the same kinds of ways of dealing with it that you find in the

philosophy of mathematics. That is to say, some people will say there IS the color

blue. It exists independently of us and abstraction allows us to get an

idea that will allow us to refer to that. That’s somebody who takes the existence of

blue apart from its instantiations quite seriously. Other philosophers might say that

you have an abstraction that gets you a mental representation that doesn’t

correspond to anything in the world, It’s just an idea. That corresponds to one

of the ways of thinking about numbers. And other people will say all you

really have are the instantiations of blue, and the word ‘blue’ or the

idea ‘blue’ is just a way of talking about those instantiations. So there’s an exact

parallel to handling the problem in the case of color to the ways you handle it

in the philosophy of mathematics. So that’s a bit of background about one way

to think about philosophy of mathematics. You get in a dilemma between giving the

objects of mathematics the ontological status they seem to demand and answering

the epistemological question of how we could have knowledge of them. So there’s

a dilemma here. And this dilemma was famously articulated by Paul Benacerraf

in an article. And one of the interesting things about that article is that with

this understanding of the dilemma, the pull on the one hand, toward giving a

metaphysically robust interpretation of what a number is and

the problem of explaining how we can have knowledge of it

gave him a way of reflecting on different schools of philosophy

of mathematics that had arisen. And it looked like the different

schools fell into these different camps: those who take numbers to be objectively

real and independent abstract objects, those who take them to be creations of our mind,

and those who think that they’re in some way manifested in the world. So we’ve

been contemplating three possible responses to the question

“What is a number?” The first answer says that numbers are

abstract objects. They’re not located in space and time, and they don’t have the

sort of causal influence on things that we perceive that things located in

space and time seem to. The second answer to the question “What is

a number?” that we’ve looked at says something like: Numbers are mental

constructs or they’re things in our minds. And then the third answer we’ve been

thinking about says that numbers are somehow in the concrete spatio-temporal

world, maybe they’re physical things even, but they’re things that we directly experience

the same way we experience chairs and tables. What would that answer look like?

I mean, would it be the number 2 is the set of all things of which there are 2?

Or how does that answer work exactly? Like so much in philosophy,

the devil’s in the details, right? It’s one thing to think you just get numbers

from the world by seeing pairs of things in the world and then

you’re home free. The difficult part is in coming up with a theory

that can really make sense of how that is to work. One thing to notice here is that those who

want to make the epistemological problem easier by locating numbers in the world

in some way make knowledge of numbers very much like my knowledge

that there’s a cup on the table here. Whereas, from the get-go, from Plato

and before Plato, there was this sense that mathematical knowledge was quite

special, that we actually do know it in a different way. I said earlier that

somebody who is committed to the idea that numbers are abstract objects is

stuck with having to give an account of how we have knowledge of them, and

that Plato and others following Plato, have resorted to talk of perception.

But of course, perception is like my perception of the cup, so it’s just a

metaphor, and they’ve had trouble spelling out the metaphor. Do we have a special faculty of perception

that allows us to know things? And Plato’s answer was yes. And that was

the answer of other people as well. One of the nice features of

biting that bullet is that it makes mathematical knowledge special in

a way that it does seem to be special. So, mathematical knowledge does not seem to

be dependent on the way the world happens to be. Mathematical knowledge also comes

with what seems to be a necessity that it be true. It isn’t just that

3 + 5=8 happens to be true, in the way it can be that it

happens to be the case that there’s 2 cups next to me

on the table right now. But it seems like it MUST be the case.

There’s a force of necessity there. And it’s also the case that mathematical knowledge, if you think

through things carefully, has a level of certainty that comes with it that has

made a deep impression on people. In fact it made such a deep impression that it’s

shaped all of philosophy. Mathematics provides a paradigm of the best kind of

knowledge we seem to be capable of. It makes us want to know everything or as

much as possible with that same level of certainty. So, this is a very interesting feature

of mathematical knowledge and it was again, I want to emphasize, deeply

influential in the history of philosophy. There’s a special kind of knowledge.

Now, go back to where we were. If you’re going to explain mathematical

knowledge as being based on the things that we experience in the world, you’re

making mathematical knowledge very much like my knowledge of the cup on the

table; and it’s no longer clear how you could explain, for example, that

my knowledge that 2 + 3=5 is actually a knowledge of a

necessary truth. How would you get that from experience by seeing apples

when you were a child or something? There seems to be something very special here.

The justification of mathematical knowledge does not seem to be dependent

on experience, it seems that the propositions of mathematics are necessary, and it seems that we can be more

certain of mathematical claims than a lot or most or all claims that we get

through our senses about the world. So that’s already an issue that’s on

the table once you take this route. John Stuart Mill is one of the most famous

proponents of the idea that you don’t need to have any kind of special faculty

of knowledge to have mathematical knowledge, you get it as a child by

seeing that 2 apples together with 3 apples equals 5 apples. John Stuart

Mill then has the problem of explaining why mathematical knowledge seems to be

knowledge of necessity. And his claim is that you’re not having insight into a

necessary truth, it’s just that you experienced things coming together in

groups in that way with such incredible consistency that it feels necessary

to you. But really it’s of the status of a claim like “there are two cups on the

table right now”. But that’s actually a challenge for the whole approach.

You’re gonna have to explain the apparent necessity of mathematical claims

in a way that seems convincing. Another challenge is really to get an

explanation of how we get from seeing 2 cups on the table to the number 2

in a way that doesn’t already presuppose the number 2, right? It’s very easy to say

that you could see 2 cups on the table and then think of the “two-ness” or

something like that. It’s not so clear that you could really do that unless you

already had the concept of 2 to bring to the table, both figuratively and

literally in this case. So you have to be very careful of how you describe how

if you’re being a strict empiricist and thinking that all your knowledge comes

from the world, how this account is supposed to work. Now, one thing you

might say is, well, there’s this two-ness of the cups here and there’s a two-ness of

the glasses on the table–the sunglasses and the reading glasses–and then there’s

what those two groups have in common. And now you are abstracting out a

representation. A common way of thinking about that is that I’m seeing the set of

2 here and the set of 2 here. But now we have to be very careful because,

remember, that the whole point of this exercise was that you were in

perceptual contact with 2 things here and 2 things there. But

we’ve already made a shift to talking about the SET of 2 here and the set of 2

here. Why is that problematic? On most accounts, a set is an abstract object,

every bit as abstract as the numbers that we were trying to explain in

the first place. So the lesson of this is once you start digging into it, giving an

account of how we know mathematical truths based on experience doesn’t seem

to be as straightforward as you might’ve at first thought. You have to account

for the necessity, the apparent independence of justification on particular

sense experiences. And you can’t, so to speak, cheat and import abstract concepts when

you’re giving an account of how we get the number concepts themselves.

That makes the project interesting, difficult, but interesting. It isn’t fatal to it.

And in fact, there are proponents of all three positions that I described earlier.

I should say though about this view that numbers are somehow manifested or

physical in the world and exist in the world; I’ve given the example of collections

of things in the world and talked about John Stuart Mill, but in fairness,

I should say that there’s a much more sophisticated version of this view

that developed at the beginning of the last century called “Formalism”.

And Formalism attempted to account for mathematical knowledge by interpreting

it as rules for the manipulation of physical symbols on a piece of paper.

That gives Formalism more resources than the view we were talking about earlier

with respect to John Stuart Mill. But it is important to see that some of the

motivations and the, so to speak, location in the classification of possible

responses to the dilemma of mathematical knowledge and the status of

mathematical objects is in that camp still. Another influential approach to the

question “How do we come to know facts about numbers and how do we come to

have knowledge of arithmetic?” is to say that we have knowledge of numbers in whatever

way it is that we have knowledge of logic; of what implies what,

of what follows from what. And this school of thought has been

given the name Logicism. So how does the Logicist view work exactly? Well that’s a very good question, an

important historical development that did impact, deeply impact, our views on the

nature of mathematics, both the nature of mathematical objects and our knowledge

of them. So, as you’re pointing out, Logicism was a project that began at the end

of the 19th century to show that mathematics is really nothing more

than logic. Now, one could have various motivations for doing this, but I could

just say quickly that if you were worried about abstract objects and it turned out

that mathematical knowledge was just logical knowledge of a special form,

you might feel better about the status of mathematics because logic does not

appear to assert the existence of objects. Also, if you were concerned about

the nature of mathematical knowledge insofar as you’re interested in how it

can be justified, and it turned out that mathematics really was based on logic,

then the justification of mathematical truths would just rest on logic. Now, in a

certain way, that just pushes everything back to logic, but it would tell you

something very interesting about mathematics and will tell you something

very interesting about logic, namely you can get mathematics out of logic and

mathematics is nothing more than logic, and so on. This raises a host of issues

even if Logicism were successful. However, Logicism was only partially

successful in the sense that if the project was to show that mathematics

is nothing but logic, that was not a successful project. What was shown was

that mathematics is something that you can obtain from logic plus set theory.

Now recall that a few moments ago I was pointing out that sets are abstract

objects. So you might have all the same worries and concerns that you had before

Logicism was on the table. And in fact, the introduction of set theory really

throws a twist into the whole Logicist project for more than one reason. First of all, set theory itself has

developed into its own field, but set theory appears to be committed to

an awful lot of abstract objects because you can build sets out of sets and so on

and so forth; or anywhere there is a set you might think there’s a set of those

sets. And very quickly you get beyond uncountably many sets and way beyond

that as well. So suddenly, what looked like a very neat reduction of

mathematics to logic seems to depend on something that is immensely rich,

powerful, and clogs the universe with infinite numbers of sets that a lot of

people aren’t too comfortable with. So what looked like a nice streamlined

reduction of mathematics to logic turns out to involve set theory which comes

with its own commitments, its own theory and a tremendous number of

abstract objects all on its own. One thing that is important to

understand is that Logicism arose at the end of the 19th century after a

period in which mathematics was a arthimetized. What do we mean by that? Prior to the 19th century, there was the

study of geometry and then the study of mathematics more generally, and there

wasn’t a unified account of mathematics. Also, very importantly, mathematical

proofs and mathematical methods appeared to rely on intuition taken in a very

broad sense, that is to say, geometrical proofs seem to require that you do

constructions and that you have some sort of insight — again I’m using

intuition in a very broad sense here. Mathematics hadn’t achieved all of the

rigor that it has today, yet there was a push for rigor, in particular, in the

foundations of calculus and that kept pushing mathematicians to think more

and more rigorously about mathematics and try to distinguish good proofs from bad

proofs, and that motivated a lot of mathematicians to try to get rid of any

kind of apparent appeal to intuition and have as strict as possible proofs. All of

that push toward what was called the arithmetization of mathematics in which

you have a theory of numbers that begins with the natural numbers and then

you explain how to, so to speak, build rational numbers or account for rational

numbers in terms of the natural numbers, then you extend also to

negative numbers and so on. Once you do that, you have now made it

possible to try to think of geometry as simply an application of mathematics.

And that gives a priority to numbers and the natural numbers in particular over

all other branches of mathematics, geometry included. That is what I mean

by the arithmetization of mathematics. There was a big drive, as I said, to get

rid of any appeal to intuition and this broad sense and to reduce everything

to logically rigorous proofs. A lot of that was not formally articulated until

the end of the 19th century when a philosopher named Gottlob Frege

develop symbolic logic of a form that we know it today. It was subsequent to the

arithmetization of mathematics that certain philosophers and mathematicians

such as Frege started thinking about what lay at the foundation of the

natural numbers themselves, and led them to the Logicist thesis. The point is that

Logicism could be seen as part of this long path of getting rid of any reference to

intuition at all in mathematics. And if you think that logic is simply a matter of

conceptual knowledge, you have given an account of mathematical knowledge that

does away with any kind of intuition or anything that could give you false

results. That was the ideal that was appealed to here. Given that Logicism

wasn’t successful, given that it depends explicitly on set theory, that gave a lot

of philosophers of mathematics pause about the Logicist project and how we

know mathematical truths after all. Is our mathematical knowledge really just

logical knowledge based on concepts or does it go beyond it in ways that depend

on more than conceptual representation? Now, even independently of Logicism,

but in particular, because of some of the difficulties that it faced, there were

philosophers of mathematics who thought that there must be some role for

intuition in mathematics; again, intuition understood in a broad sense. And Henri

Poincaré was a famous example of this. He thought that mathematics required some

sort of mathematical intuition. Those who are inclined to agree with Poincaré, in one

form or another, often look back to Kant because Kant had insisted that our

mathematical knowledge was not simply conceptual, but required also intuition. Now what’s important to understand here

is that when Kant said ‘intuition’ he wasn’t just talking about some quasi-

perceptual insight into facts or something like that, he had a very

precise notion of intuition that contrasted with concepts. Both

concepts and intuitions are kinds of representation that we have. Conceptual

representation is distinguished from intuitive representation in that the

latter has a, what I will call, quasi- perceptual format. That is to say,

it’s not the kind of thing that you think paradigmatically fits into the form of a sentence where you

break things down into subject and predicate and make a simple assertion

like “the cat is on the mat”. To give an example of what Kant had in mind,

an intuition is like a spatial representation in which you have the

representation of parts next to each other located in space, and that’s a very

different kind of representation than a concept. Kant claimed that all knowledge

requires both concepts and intuitions to work together. Those who recognize

the limitations of Logicism or were in fact opposed to Logicism look back to

Kant to understand why he thought intuition was required in mathematical

cognition and mathematical knowledge. Now I said that Kant says that intuitions

are quasi-perceptual, the reason I say quasi-perceptual is that he doesn’t

think that they are perceptual in the way that most perception is empirical,

the way things [like] my perception of the cup on the table is empirical. Rather,

Kant wanted to allow that some kinds of quasi-perceptual representation are

pure and a priori. Notice that this would make mathematical knowledge very

special and different from my knowledge of the cup on the table, but at the same

time, it has something in common with it because it has some of the features

of perception, the way spatial representations are perceptually

represented for example. The big question then was for those who are looking

back to Kant and are interested in getting at the bottom of Kant’s views is “How

does intuition play into our mathematical knowledge in the sense of intuition that

Kant is interested in?”. So there is still a very live question today, 200 years

after Kant, about what is the nature of mathematical cognition, where the form

of the question is what sorts of concepts or quasi-perceptual representations are

presupposed by mathematical knowledge. And that means that Kant’s views are still

of very much of interest today because even though mathematics is developed

tremendously since Kant’s time, we still do not have a clear theory about

how we have mathematical knowledge. Daniel Sutherland, thank you very much for

joining us. Thank you, thanks for having me.

First!

John V. Karavitis

Whether the "number 2", or 2 items, they seem both to be just abstractions… But for the sake of utility, they are necessary abstractions. To say there are "two apples" assumes that our definition of "apple" is complete and coherent. One could argue that an "apple" is just a label we have created, an abstract label. But that abstraction is a necessary abstraction, just for getting along in the world. In the end, it could be a false label (or at least partly false), but taking such a "false" view was indeed helpful until we get to a fuller view of "the truth". Moral of the story? Let's not get too intense in our "will to truth". Lying/abstractions/etc can sometimes be helpful to getting to a new plane.

👍

I have become a megafan of George Lakoff. He explains our knowledge of mathematics as built upon metaphors of the type "a number is a collection of objects", or "a number is the length of a segment". These in turn are built on top of our innate abilities to subitize (tell how many objects there are in a collection without counting, only works up to about 4 or 5), separate the world into objects etc. From what I understood of Kant's description of intuitions, these innate abilities might be the type of thing that he was talking about, this would leave up to cognitive scientists to discover what they are by finding out how the brain works.

In Lakoff's view, any property that we assign to the abstract concept of number is a property of the real, concrete, world (like putting together a collection of objects gives the same result no matter the order in which you do it gives rise to the commutative property of addition).

Doesn't the first guy who speaks also speak on The Partially Examined Life?

twice I lost a long explanation of numbers.

and my tablet froze. now I'll make it short.

we invent words . words are representative or abstract. numbers are words symbolized. also abstract.

math is a word problem in symbolic form. example. 2. two. 3. three. and so on.

Kant touch this

It seems to me that the empiricist has a problem when it comes to higher mathematics. Sure, we might be able to define the number two from the experience of seeing two things, but mathematics is also about infinities, non-Euclidean geometry etc., which aren't exactly things we bump into in the same way as we bump into 3+5 apples = 8 apples.

How would Mill answer to this problem?

I think that words and numbers are abstracts both, the meaning give them perceptions in our minds, mathematics is real precision leading to perfection, but if we admit that mathematic come from nature arround us, then maybe nature is perfection, what is mathematic if it doesn't include nature.

People are absolutely worried about whether or not colors are parts of objects, or whether we see them in "the mind." There is a whole philosophy around this called the philosophy of color- great works by Hardin and Hurvich- all have studied the problem of these secondary qualities as Locke called them. Paradigms about the idea of whether color is abstract or concrete have been called subjectivism i.e., projectivism and objectivism. Notable philosophers have designed theories about this problem – like Marr in his "Vision" (1982).

The interviewer might have meant something different but it comes off completely misinformed sounding.

If you just think about it commonsensically- one could easily worry about the idea of color being a part of objects, or in the mind, just given the notion that one cannot transcend another's subjective experience in order to know whether they're blue is the same as the others' blue. Theoretical and commonsense perspectives can both be taken (and have ensured historical debate and dialogue) on the ideas of perceived qualities and the nature of them. It's not just mathematics but abstracta in general that is cause for philosophical and scientific concern.

Great breakdown!

What about that there is when the numbers don't appear in reality? Does that mean we haven't found them or what? I'm loosely talking about math for the quantum field and they are not what quantum & regular physics tell us to be. Or is that random built in the fabric of reality & energy??? I'm high lol

I enjoyed this video very much.

I listened 5 minutes and I lost myself. Is there someone who is able to write down the gist of this 30 minute speech? Many thanks in advance!

Absolutely none of which addresses the question that one starts with matching (say) sheep against stones to see if the flock has decreased over the day, which doesn't even require counting, yet wind up withe Godel's theorem as implicit in the development of 'matching' into 'counting.' From' where?

Clearly, we've made no progress on the philosophical side. Equally clearly, we don't need to understand what we're doing, to do it very well. Math, musical composition, writing … this is true of all of them.

Numbers are the names of the properties by which objects are distinguished.

Good day, fellow nerds!

Maths is a phenomenon caused by pattern in nature, patterns that can extrapolated into rules. Nature is not random chaotic thing, otherwise it would not exist

Numbers are labels we slap on patterns to sort them out!

The key is in Immanuel Kant's "Critiquie of Pure Reason". The maths are based on the apriori axioms. The aprioris are the engine of Reasoning. It is not possible for humans to discern if the aprioris are some kind of truth separate from human experience, and in fact they are necessary for the human experience of Reason. Fortunately the aprioris have been very useful for improving the ease and enjoyment of life for humans. Ultimate truths are not something humans can discover.

Why these idiots can't figure out these language vs reality and abstraction is because they still do not have a handle on the structure of the human neocortex.

How about: First we realize that groups of 2 and 3 things spur the concepts of the numbers 2 and 3. Then we look at physics problems, such as gavity and the orbits of planets and find more evidence of an underlying mathematical essence. So, this is all pointing to a fundamental essence of the world: mathematical concepts. Not the other way around.

I am at the minute 10, but I don't see why compare colors with numbers.. after all, colors are indeed perceive, we SEE them. I don't see a number 2 in the air (!??)

Physicists will ignore my points for selfpreservation purposes but you won't… Check on my thoughts at https://docs.google.com/document/d/13DYbeBfq1m-LnCDrs4CWUc2S0n6pv5vluCM4il5Js14/edit?usp=drivesdk

https://docs.google.com/document/d/1xtzuM96VWg-QwVpxpK2Bbq_VjjyW3nWhIuoh02Tsd8Q/edit?usp=drivesdk

But if at a certain point physical conceptual math become abstract then isn't sugondese now not logical?

Wonderful podcast, this is my first podcast (I heard there are educational ones out there) and it was definitely worth it, thanks

I’m assuming this is about Platonism but the video just started.

… still one of my favorite philosophical subjects – I never tire of thinking about the nature of number 🙂

The colour blue exists in white light, therefore, it exists independently of the things it is instantiated in. You could say it originally comes from white light which contains the whole visible spectrum. It is, in that sense, a manifestation of a certain aspect of the whole.

No causal influence?! That's crazy! Numbers are, at their most primitive root, a way to quantize our notion of amount. Place two bowls of food in front of a dog – one completely full and one only a quarter full – the dog will go for the bowl with more food because even dogs have a notion of quantity capable of causal influence. Once we have a notion of quantity we just need to define what it is to have something and what it is to have nothing and then use Peano's axioms and we have the natural numbers.

Excellent introduction to Philo of Math. Speaking for myself, I’m a late Scholastic Realist (Duns Scotus) who believes whatever is in whatever way is real. In what way real is what demands investigation. The verb “to exist” is the trouble maker here. We associate existence with space and time – ultimately with matter (pace Descartes). Peirce is a Scholastic Realist who offers another solution path, namely his semiotic (sign theory) that is thoroughly anti-Cartesian dualism (mind/body dualism). Numbers and other non-physical realities are given due justice and equal standing among whatever is in whatever way. Great talk. Thank You!!

I seem to fall in the category of people who believe we have a special perception for numbers. It fits with the notion that we are creatures evolved in this world, that have great utility from building models of the world, and imagining modified versions of the world, that we can desire to bring it to, and the motivation to action of this desire, is called the will of man. Plato thought the static examination of our perceptions was the highest ideal, but Nietzsche believed the will was a higher ideal. This is a very interesting dichotomy between two ideals, because one can see the will to knowledge as being a subcase of the Nietzchean ideal, and the will only being breathed into life, through the logos, by imagining, and seeing an avenue for a possibility, and so we have the partial converse, that the conscious will is dependent upon the logos.

But the remarkable insight by Nietzche, that the beautiful is what produces the arousal of the will, completes the converse statement. That is, is by bringing our model of the world to the conscious mind, we may witness beauty, and thus the activation of the will.

I don't understand the comparison of colors and numbers. Humans have invented multiple numbers and other objects over time. We just make them up by defining a set and some rules, and then start exploring their properties, relationships and the structure that arises from the exercise. It is actually easy to invent numbers. But I still haven't heard of anybody inventing a color (giving an arbitrary wavelength a name is not inventing a color, is just taging a measurement).

It surprises me that anyone should 'THINK' numbers are real. Here's my conclusion after years of questioning:

As with words, 'numbers' are merely conceptual labels. And, like words, numbers represent a predetermined meaning, a 'value' that we attach to real or imagined objects.

In the same way that a sentence is the mental processing of words, mathematics is also purely, the MENTAL PROCESSING of numbers.

Therefore, it should be self-evident that universe cannot possibly have been 'created' by a MENTAL PROCESS…

…Or could it?

and guess how will this all end – after centuries or even millenia of fruitless thrashtalk by all these "philosophers" (called theologians before) some random scientists come and simply explain stuff… taking literally EVERYTHING from the parasitic dirty hands of these phlidiots (who nowadays care only about their parasitic "pseudojobs" at the pseudouniversities) – as it happened with math, physics, chemistry, biology… and when they lost the last bastion (natural language analysis), they are hanging by a thread in this pseudoscientific realm of "consciousness", especially degenerated in that chalmersian mystification "hard problem" (yes, you can have all these thousands of "jobs", you can spend money via crime called STATE doing all these wonderful "conferences", writing "books"…

Albert Einstein On Himself: "Do not worry about your problems with mathematics, I assure you mine are far greater."

On Life: "A table, a chair, a bowl of fruit, and a violin; what else does a man need to be happy."

Etc. ~ Albert Einstein Quotes

There are also mathematical entities that cannot be known by experience… null, infinity, etc…

Sometimes I feel like most of philosophy of mathematics literature is written by people who can’t factorise a quadratic… it’s always talk of two apples and three apples…it’s quite embarrassing. For example, quadratics (solvable by integers) are intricately related to prime numbers. This isn’t I see apples set theory blah blah blah waste of time analytic crap.

Numbering numbers = time duration timing, positioning-identity, in the dynamic shape shifting universe of quantized balanced constants, ..such that multiples of e-Pi-i Eternity-now Singularity Infinity are the formulation of one probable continuous creation connection in symmetrical balance reciprocals, in this Universal Mathematical Principle.

Is philosphical inquiry possible in a youtube comment section? No. How should one view independent philosophy’s utility? Get your money up first bb