# Intro to the Philosophy of Mathematics (Ray Monk)

I'm going to talk about, give a general

introduction to, the philosophy of mathematics. And I'm not going to assume

any knowledge at all of either philosophy or mathematics. So I'm sorry

if it seems a bit basic to some of you. Okay. I mean, the first thing to say, I

think, is that mathematics and philosophy which I think most people tend to think

are poles apart, are actually closer than you might think. And a lot of the great

philosophers have also been great mathematicians. And I think that's not

an accident. I think there's a close affinity actually between the two subjects in

several respects, the most noticeable perhaps is that both require and

demand thinking on a very abstract level. But as I hope to show today, there's a lot

more to the relationship between the two than that, and in particular, that

mathematics has furnished philosophy both with a model of a certain kind of

knowledge and with a set of deep and interesting philosophical problems.

But the two go together. Plato knew this. It's said that at

the opening of Plato's, at the gate of Plato's Academy, it said: "let no one

into here who knows no geometry." Interesting, I think, that it's

geometry rather than arithmetic. And there's a story there which I'm

going to tell which is to do with the fact that the ancient Greeks regarded

geometry rather than arithmetic as the more foundational, the superior

branch of mathematics. And one of the main reasons for that was the unhappy

story of Pythagoras. Most of you I think will know the name Pythagoras from his

famous theorem which we all learned at school which says that the square on

the hypotenuse of a right angle triangle is equal to the sum of the

squares on the other two sides. Well, that very theorem presented a

problem to Pythagoras and his followers. I don't know whether many of you know this

but Pythagoras was a sort of cult figure in ancient Greece, and he had a band

of followers who were dedicated to the mysticism of numbers. It was one of their precepts that

everything in the world could be expressed as number, and in particular,

as the ratio between two whole numbers. Well now, if you go back

to Pythagoras's theorem, it's a consequence of that theorem

actually that that belief is unsustainable. And the reason for that is

that reflecting upon that theorem leads you straight into what are called

irrational numbers, numbers that cannot be expressed as the ratio between two

whole numbers. To see this, imagine a right angle triangle with length 1 here

and length 1 here. So you've got a square on here of 1, a square on here of 1.

The square on the hypotenuse therefore is going to be the square root of 2.

And Pythagoras discovered to his horror that the square root of 2 cannot be

expressed as a ratio of two numbers. It's not a rational number. Pythagoras was so horrified by this he

swore his followers to secrecy about it. They weren't allowed to mention

the irrationality of the square root of 2. But it did undermine the faith in

numbers as the foundational view of mathematics. And that's why geometry is

regarded as superior because in geometry you can express the hypotenuse,

you can draw it. But in numbers, you can't express it as a ratio. Okay. So,

mathematics and philosophy have always gone hand-in-hand. I said one reason

for that is that mathematics provides a model of knowledge. It provides a

model of knowledge of a particular kind. That's to say, philosophers since the

ancient Greeks, for over 2,000 years, have regarded mathematical knowledge as

somehow special and something to which all knowledge could possibly aspire. What

makes mathematical knowledge special? Well, several things. One is, unlike other knowledge, it's certain.

If something is true in mathematics and if you know it, you're not going to doubt it.

You're not going to say, "well, I'm fairly sure that 2 + 3 = 5", you know that 2 + 3 = 5

with absolute certainty. That's the first thing. The second thing is

that knowledge in mathematics seems to be incorrigible, that's to say, it's not

going to be corrected by anything you might subsequently learn. 2 + 3 = 5

now, 2 + 3 = 5 two thousand years ago, and you might say, even if

there weren't any people on earth, it would still be the case that if 2 apples

fell to the ground and 3 more apples fell to the ground, then you would have

5 apples on the ground altogether. So mathematical knowledge is certain, it's incorrigible. Another thing is, that's

related to its incorrigibility, it's eternal. It's always true. If something is

true in mathematics, it's not true for the time being. It's always true and

always has been true. And a fourth thing about mathematical knowledge is that it

seems that mathematical truths seem to be not just contingently true,

but necessarily true. Southampton happens to be on the coast

of England. But that's not a necessary truth, it doesn't have to be. I happen to be wearing black trousers.

But again, that's not necessary. But 2 + 3 could not equal anything

but 5, it's necessarily equal to 5. So, philosophers have looked at this example

of mathematical knowledge and they've thought two things, one is: How is it that

knowledge in mathematics has those characteristics? What is it about

mathematics that gives its knowledge those characteristics? And the

second thing that's occurred to several philosophers including Plato, Hobbes,

Russell was: Why can't other kinds of knowledge be like that?

Maybe we could have a system of physics, for example,

that made our knowledge of physics as incorrigible, eternal, certain & so on

as our knowledge of mathematics. So, mathematics has provided a

model of knowledge. On the other hand, it's inherently puzzling. Mathematics

has provided philosophers with a number of deep puzzles that we have been

scratching our heads over for over 2,000 years. The central one of which, is the

most basic of which, which is: What is mathematics about? So, if

you know that 2 + 3 =5, you know with certainty, incorrigibility, and so on. But

what is it that you know something about? As I said if you add 3 apples to 2

apples, you'll get 5 apples. If you add 3 lemons to 2 lemons, you'll

get 5 lemons. But 2 + 3 = 5 is not about apples, and it's not about

lemons. It's applicable to those things, but it's not about those things. So, what is it about? Well, it's about

numbers. But what are numbers? It's when you ask that question–it's one of those

philosophical questions that, the more you think about it, the less clear it gets.

What are numbers? Numbers seem to be the content of objective truths. If I say

that 2 + 3 = 5, I haven't just made it up, I'm not getting it from anybody. It is

objectively true. And most of the things that we have objective truths of

are objects. So are numbers objects? Well, unlike apples and lemons, you can't

see a number, a number does not reflect light. You can't smell a number, you can't

touch a number. So if a number is an object it's a peculiar kind of object.

It doesn't exist in space and time, it doesn't corrode,

it doesn't get old, it makes no sense to ask what its

physical size is and so on. So if it is an object, it's a peculiar kind of object.

It's an object that doesn't exist in space and time. It's not part of our spatial-

temporal world. Plato had a theory to account for this, which is

the famous theory of Forms. According to Plato, numbers are Forms, and

Forms are abstract, objectively existing objects. And the fact that they're not

spatio-temporal, they're not part of our world of space and time, didn't

bother Plato a bit. On the contrary, it confirmed him in his opinion that

reality is formal. The world of Forms is the reality of which our spatio-temporal

world is but a shadow, according to Plato. And according to Plato,

this accounts for the corrigibility of everything we get from our senses–

everything we see, everything we touch, everything we smell–the knowledge we get

from our senses is always open to revision. We might look again and

see something different. Whereas, formal knowledge is not. And

what that shows, according to Plato, is the superiority of our reason over our senses.

We can't see Forms, we can't touch Forms, but we can grasp them intellectually.

We can get knowledge of Forms, knowledge of arithmetic by

thinking. And according to Plato, that explains the characteristics of

mathematical truth: it's necessarily true, it's not contingently true. It's not

contingently true because it's not about the contingent world, the spatio-temporal

world. Now, the downside of this theory —the upside of this theory is that it

explains a lot about mathematical truth and mathematical knowledge. The

downside is that it requires us to believe in something that a lot of us

have trouble believing in, which is: an objectively existing world of Forms. And even if we could persuade

ourselves that such a world existed, we'd have an apparently insolvable problem

which is: How do we, so to speak, reach it? Given that we can't see it,

we have no sensory awareness of it, how do we bridge the apparently

unbridgeable gulf between that world and us? After all, we do exist in the spatio-

temporal world. So, for those reasons, a lot of philosophers,

I would say most philosophers, have had trouble persuading

themselves that Plato's world of Forms really exists. And therefore, that the

so-called mathematical realm that was supposed to be part of the world of

Forms, a lot of philosophers have trouble persuading themselves that that exists

as well. And these thoughts had occurred to Aristotle who was a pupil of Plato's,

who didn't believe in the world of Forms. He believed that mathematics is not

about objects in the mathematical realm, there is no such realm. Mathematics is about our world.

And okay, we can't see a number, but we might regard a number as

a property of things that we can see. So okay, numbers are not objects, but we

can understand them as features of objects. So we look at a field,

we see 4 cows, it's not that we see the cows and

then we see 4. It's that 4 is a property of that collection of cows that we see.

Well, philosophers since Aristotle have put forward powerful objections

to that way of looking at numbers, the most powerful of which were put by a

German mathematician-come-philosopher, Gottlob Frege, who was writing in

the 19th century. And he said look, when we know something about numbers,

we know it objectively. But numbers, Frege said, cannot objectively be

properties of other things. And the reason he said for that is that which

number belongs to a collection of things will depend upon how we conceptualize it.

So think of a deck of cards. A deck of cards has 52 cards in it. It has 4 kings,

it has 4 suits; 4 suits, 52 cards. Depending on whether—so we

have a deck of cards in front of us— depending on whether we're thinking

in terms of cards or of suits of cards, different numbers will belong to

that particular collection of things. So, does that collection of things have the

property 52? Or does it have the property 4? It has the property 52

if we're thinking of cards. It has the property 4

if we're thinking of suits. At a simpler level,

imagine a pair of shoes. It's 1 pair of shoes, but 2 shoes.

So, as an object, as a physical object, which number belongs to that? Is it the number

one or the number two? So, Frege says this is in general true that objects,

objectively, so to speak, do not have numbers as properties.

They acquire numbers as properties when we think of them in different ways.

And this is inconsistent with regarding mathematics as objective. So for those

reasons, the idea that numbers are properties of objects is one of those

ideas in philosophy that is, by a lot of philosophers, been regarded

as being refuted, refuted by Frege. We seem to be on the horns of a dilemma

where, when we solve certain problems about mathematics we encounter others.

If we do justice to the objectivity of mathematics, like Plato did, we seem lumbered with a kind of metaphysics,

a metaphysics of abstract objects. And when we asked ourselves

too deeply questions about what these abstract objects are supposed

to be and what the world of Forms is supposed to be, we find that we can't

give satisfactory answers to those questions. Where we can give satisfactory

answers to those questions we seem to be impaled on the other horn, which is,

we seem to have adopted a view which does away with the

objectivity of mathematics. Okay so, fast-forward now from the

Ancient world to the 18th century, Europe in the 18th century, and you come across

a great towering figure in philosophy, the German Immanuel Kant, whose great

work was the Critique of Pure Reason, published in the 1790s. Kant put forward

a theory of mathematics which became the most influential up until the 20th

century. And Kant did so in a way that grasped, as it were, one horn of this

dilemma, and put forward a theory that did away with the objectivity of

mathematics. Kant's thinking about mathematics starts with a question that

I didn't raise about mathematical knowledge but which is implicit in some of the

things I did say about it, which is this: If something is true mathematically, it's

necessarily true. And yet, mathematics works. 3 apples plus 2 apples really

is 5 apples. The world, as it were, seems to conform to the laws of

arithmetic. But the laws of arithmetic are not just true, they're necessarily true.

And Kant's question was: "How can we know something about the world which

is necessarily true?" So he introduced two distinctions which have since become

part of the technical vocabulary of philosophy. The first distinguishes two kinds of

sentence: an analytically true sentence and a synthetically true sentence. The difference is this: an analytically

true sentence is necessarily true. So, for example, "All bachelors are

unmarried". That's an analytic statement. It's an analytic statement because

it's true by definition. Compare it with the statement "All bachelors are unhappy".

It might be true that all bachelors are unhappy, but it's not necessarily true, it's not

part of the definition of a bachelor that bachelor is unhappy. But it is part of

the definition that a bachelor is unmarried. So the statement "All bachelors

are unmarried" is necessarily true because it's true by definition.

Kant called this an analytic statement as opposed to a synthetic statement

such as "All bachelors are unhappy". The reason he chose those particular terms

"analytic" and "synthetic" is to do with the question of whether you're dealing with

one concept or two. The idea here is that if you say "All bachelors are unhappy", you're making a synthesis of two quite

different concepts: the concept of being a bachelor & the concept of being unhappy.

If you say "All bachelors are unmarried", you're not synthesizing two

unrelated concepts, you're analyzing, so to speak, a feature of one concept. It's a

feature that — you know, what does the word 'bachelor' mean? It means unmarried man.

So if you analyze the concept 'bachelor', you can analyze it into the concepts

'unmarried' and 'male'. So that's why he said that's an analytic statement as

opposed to a synthetic statement. So there's another distinction he

drew which is in regard to how we know things to be true, and he used Latin

titles for this: 'a priori' and 'a posteriori'. Something is a priori known

if our knowledge of it is prior to any experience, any observation,

any testing. So again, we know that all bachelors are unmarried, we don't have

to do a survey, we don't have to do a test. We know that a priori, we know that prior

to any testing, any surveying, and whatever. Whereas, "All bachelors are

unhappy", if it's true at all, it's going to be true a posteriori, it's going to

be true on the basis of doing some empirical research. We know

that smoking causes cancer, we didn't always know that, but we do

know that now. Why do we know it? Because we've done experiments, we've done

tests, we've looked at people who smoke, we've made observations, and so on.

So "smoking causes cancer" is a posteriori. Now, think of those two

distinctions. It ought to be the case that the analytic goes with the a priori,

and the synthetic goes with the a posteriori. But what Kant said about mathematics

— and this is to do with it being necessarily true and true of the world —

is he says in mathematics we've got this curious hybrid. He said mathematics

is not analytically true, it's not true by definition that 5 + 7 = 12.

So that's a synthetic statement according to Kant. And yet it's a priori.

We don't have to do any experiments to find it out. So mathematics — and this,

according to Kant, is its great single feature — mathematics is synthetic a priori.

And his question was: How on earth how can we know things "synthetic a priori"?

And his answer was the whole system of metaphysics that he puts in the Critique

of Pure Reason, which is called "Transcendental Idealism", at the heart of

which is the idea that we don't know anything and cannot know anything

about things in themselves. We can only know–so he called those

things 'noumena'. We can only know things about what Kant called 'phenomena',

which are not things as they are in themselves but things as they appear to us. Things

as they appear to us, according to Kant, have been put through a kind of filter

which is the way we see the world. And mathematics, according to Kant,

is that filter. In other words, we don't get mathematics from the world,

we bring it to the world. So if you think of mathematics

as divided into two parts: geometry and arithmetic. Geometry, according to

Kant, is the spatial form through which we see the world. It's the spatial glasses,

as it were, that we look at the world. The world appears to us to be

three-dimensional Euclidean space. It's the world, the space described by the

system of geometry that we got all those thousands of years ago from the ancient

Greek geometer Euclid. And the reason those things are necessarily true,

they're true a priori according to Kant, is that we didn't get them from the

world, we brought them to the world. We look at the world

through those spectacles. Where does arithmetic come in? Where

do numbers come in? According to Kant, just as geometry is the form of our

spatial awareness, our spatial intuitions, arithmetic is the form of our temporal

intuitions. So at the heart of arithmetic is a sequence of numbers 1, 2, 3, 4, 5… It's a one-dimensional sequence, which

corresponds, according to Kant, to our experience of time. We experience space

as three-dimensional. We experience time as a one-dimensional sequence,

which is the sequence of numbers. So numbers are the way we organize time

one moment after another moment. Geometry is the way we organize space.

And put together, they give us the framework of the spatio-temporal world that we

experience. Everything we know is known about the spatio-temporal world. In other

words, everything we know, we know through those spectacles which we

ourselves have put to the world. Okay, well I've said that was enormously

influential theory of mathematics. It was opposed in the late 19th and early 20th

century by the man I mentioned earlier, Gottlob Frege & by a British philosopher

and mathematician, Bertrand Russell. What Frege and Russell together

sought to do was replace the Kantian view of mathematics with a view

that did justice to the objectivity of mathematics. And they went

right back to Plato, with this proviso: that they reverse the ancient Greek

priority about geometry and arithmetic. The ancient Greeks regarded geometry

as foundational, Russell and Frege regarded arithmetic as foundational.

And this is for two reasons. One is in the middle of the 19th century, alternatives to

Euclid's system of geometry were discovered: Riemann's system and Lobachevsky's

system. And what these systems did is they dropped the assumption

that parallel lines will never meet. In these systems of geometry, parallel

lines do meet. And what that means is that in these systems, space is curved. If you

think of the space of the outside of a globe, think of drawing two parallel lines,

they're going to meet at the top and meet at the bottom. So in these systems,

parallel lines can meet, which means that space is curved. And what really

threw the cat among the pigeons– it was bad enough for Kant's theory

that there were alternatives to Euclid because now the question arises: "well,

which pair of glasses should we wear?" And according to Kant's theory, we

shouldn't have a choice about that. But the cat was really put among the

pigeons with Einstein in his theory of relativity, according to which physical space

is Riemannian and not Euclidean. It's not just that you can — I mean,

Riemann in the middle of the 19th century invented the system of geometry,

as it were, just for the hell of it because he was a pure mathematician.

He wanted to see what would happen if you drop the parallel postulate. But according

to Einstein, it's not just of theoretical interest, the world is Riemannian, physical

space is curved. Now in the light of those developments, together with a

second development, which is that it was discovered in pure mathematics that you can

build geometry upon arithmetic and algebra. And so Frege and Russell regarded

arithmetic and not geometry as the foundational branch of mathematics. So for

them the central question was about number: "What is number?" And they weren't very

happy with Kant's answer to that, which is: Number is something inside our

heads that we bring to the world. They weren't very happy with that because it

makes arithmetic about what's inside our heads. Whereas, for Frege and Russell,

it was crucial that arithmetic is a body of objective knowledge. And so they

went back to Plato, into Platonism, it's objective knowledge about forms.

Now both of them, quite separately–and this is a remarkable fact–quite separately, both

of them had the same thought about that, which is: we can do justice to Plato's

formal theory of arithmetic — the idea that arithmetic is really about things

and these things are forms — we can do justice to that if we show that

mathematics and arithmetic in particular is just logic. So their view is called

"Logicism". And it's the view that arithmetic can be shown

to be a branch of logic. Now this, of course, only leads to

Platonism if you take a Platonic view of logical objects, which is exactly what

Frege and Russell did. According to Frege and Russell, logic is about forms,

and forms really exist. How do they make that plausible? They made that plausible

by bringing together two things that previously had not been brought

together, which were logic and arithmetic. It's been a hundred years now and

over that hundred years we've become accustomed to the idea that logic and

mathematics have got strong things in common with each other. But a

hundred years ago that wasn't the case. A hundred years ago, logic belonged to

the humanities. Logic was what you learned if you learned literature, poetry,

rhetoric. It was part of the humanities. You learned Aristotle's system

of logic. Learning logic went hand-in-hand with studying classics at Oxford.

Whereas, mathematics was what you learned if you were a scientist. And they were not

considered to have very much to do with each other. Logic was to do with language. It was

to do with using language to construct arguments, and the logic of Aristotle

tells you which of those arguments are valid arguments and which are not valid

arguments. Whereas mathematics gives you techniques that you can then use in

science. Frege and Russell brought them together with a particular theory of

number. And I think I've just got time to expound this theory. This theory makes use

of the notion of a "class", a class of objects. All right, and the way you get to that

notion is this: through language. So you start — I mean what they're doing is building a

bridge between Aristotle's theory of logic and the study of languages, on the

one hand, and arithmetic on the other. And the way that bridge works is this.

You start with propositions, with sentences, with what is analyzed in logic. Okay so take some some sentences: "Plato is wise", "Aristotle is wise", "Socrates is

wise". Those sentences all have the same form. Now you could capture that form

by replacing the name with a variable, with 'x'. So now you have "x is wise".

Now, a class is this. A class is all the things that would satisfy that sentence

if you replace the x with a name. If you do that, you've got the class of

wise people. So the class would have in it Aristotle, Plato, Socrates, all those

people who could replace the x in "x is wise". So the jargon that they came up with for

this was: "Plato is wise", "Socrates is wise", those are propositions, but "x is wise"

is a propositional function. And the class is the extension of the propositional

function. The class is all those wise things. Alright so now they used that

notion to talk about numbers. Numbers are classes according to Frege

and Russell. The number 4 is the class of all those things that have 4 members.

So there are 4 points on a compass: north, south, east and west. There are 4

Beatles: John, Paul, George and Ringo. Collect together all those things that

have 4 members, and that, according to Frege and Russell, is the number 4.

Now notice that this is very different to the old property theory. It's not that the

number 4 is a property of those collections. The number 4 is an object, but

it's a particular kind of object, it's a class. And they built a whole

system of logic on that notion of class. Then, and I'm not going to go into this

because you've probably had enough of all this sort of thing, but–

I will go into it if you want, but we'll leave that for the question period. In 1901, Russell discovered a problem

with that theory of classes which is called Russell's paradox. And he sent it to Frege. He said, look I've just

read your work, your work is very similar to mine it makes use of class theory, have you thought about this problem?

He showed the contradiction that arises in the theory of classes. Russell,

at that time, was quite confident that the problem could be overcome, poor old Frege had a nervous breakdown.

And after spending hospital, he came out and said he wasn't a Logicist anymore,

he didn't believe Logicism was true. Russell persevered with it, but came up

with a different view of logic which he got from his pupil Wittgenstein,

according to which logic is not the study of objectively existing forms. According to

Wittgenstein, there aren't any forms, this is a myth. There aren't any forms. What you've got is

language and ways of putting words together. And according to the rules for putting

words together, sometimes what you end up with is what's called a tautology. So, if I say that "It's raining outside", that's gonna be either true or false.

If I say "It's not raining outside", that's gonna be either true or false. But if

I say "Either it's raining or it's not raining", that can only be true. That is a

tautology. That is necessarily true. And so Russell took this notion from

Wittgenstein and said that's what mathematical propositions are, mathematical

propositions are just tautologies. The reason they're necessarily true

is exactly the same as the reason that "All bachelors are unmarried"

is necessarily true, they're true by definition. And so

Russell towards the end of his life said that if there was a god–notoriously

he didn't believe there was a god– but if there was a god, he said, the truths

of mathematics would have exactly the same profundity as the truth that

"A four-legged animal is an animal". And on that note, I'll finish.

Thank you.

Very informative lecture and given with great examples to consider

So he's the one that wrote a biography of Bertrand Russell.

Lojban

got it, basically what you're layin' down is:

'I am the very model of a modern Major-General

I've information vegetable, animal, and mineral

I know the kings of England, and I quote the fights historical

From Marathon to Waterloo, in order categorical

I'm very well acquainted, too, with matters mathematical

I understand equations, both the simple and quadratical

About binomial theorem I am teeming with a lot o' news

With many cheerful facts about the square of the hypotenuse'

Yes.

It's always seemed obvious to me that math is simply logic using numerical symbols. It's been so bizarre learning that this hasn't been the view of so many others for so long and that there's been so much debate about it

They are confusing themselves. They do not know it, even though they have limitations in their thinking.

They try to match the ideal world of infinity on the basis of the finite physical world, and ignore or overthrow on the back the things that have been premised. They try to deny the right logic by puns and appeal to people.

They do not know that the physical world is a special case of temporal realization through imperfect matter in the ideal world of infinity. They do not know why the physical world is finite, imperfect, always changing and so passive except a inherent property. They say those are wrong if they can not prove physically or make it happen.

They do not know what physics and mathematics are fundamentally different. They do not know for sure that physics can be discussed only if it is based on matter, and that mathematics is a logic that can be discussed without presupposing matter. They do not understand why physics can not handle zero and infinity but mathematics can handle them.

Everyone is a crook.

The problem is that truth in mathematics is conditional. The statement "3 + 2 = 5" is true by definition, which entails a vicious circle.

He's not a very good speaker.

I really enjoyed that, I have deleted my comments made as the talk evolved, I am naive but it made a whole lot of sense 🙂

Is not arithmetic just part of the language used to describe the fundamental truths of Geometry?

Thank you so so much

Thank you! This is amazing

How do you know that 2+3=5?

At haha hahaha. …I stopped listening. .lol.

7+5=12. No appeal to evidence is needed to establish this truth. Just an appeal to argument. Kant is inventing problems where none exist. Mathematical truths are not established through experimentation – they are reached through argument – proof.

They confuse themselves while exploring in depth, the mathematics world which deals with always right logics (truths) and the physical world which constantly changes and is imperfect. The physical world is a constantly changing world that imperfectly realizes mathematical truth temporarily through ambiguous substances.

I don't understand the importance of the condition "if there was a god" for what Russel said, in the end of the video.

If 4 is the class of things that have 4 members, how can you say something has size of 4, since size is not something with 4 members?

And did Russell solved his paradox?

11:43 But what do you mean by "exists"?

I sympathize, but verbiage to build dam walls to hold back subconscious compulsions, without societal frameworks, is this really cause for so much concern? Consequence of millennia worth of delayed gratification that leads to psycho dynamic alkalinity and acidity in earth's atmosphere, that drives us to tear down and level society subconsciously?

Where the wild things are? Outside the intellect that is strictly controlled by empirical powers? Sure this is not an all purpose "get out of free jail" card, cum advertisement for the western viewpoint

In my willingness to not believe such a self serving agenda, issues still remain with rise of non European powers, who may soon have equal amounts of grunt and brute forces, amplified exponentially by modern powers?

Even if this threat is real, clamping and flexing only creates hard walls that only increases these volatile forces no? So ideations and scenarios that drive human energy generating, along the lines of fretting and ringing hands, when not bearing teeth and shaking fists!

Driven by belief that they are marred and disfigured inside, and thus fear those creatures that are locked in the sub basements of mind and collective consciousness. Where your fears are a threat to others

Yet I feel that the main causes for pain, damage and trauma in life is misalignment rather than any fundamental flaw. Scrapping against a thing that is incorrectly aligned is what causes damage after all.

And yet the basis for this square peg in round hole age, and the subsequent bi product of abrasion that is the core energy source of today's society, is it not that, that has led to toxicity and build up of anxieties, aggravation, desperation.

That then powers more ruthlessness and cruelty? Knowing the self, it's always better to be preemptive rather than letting a rival for resources and control, use the deep evil (powered by fear and desire and then furthered by selfishness) we all have inside of us.

Now that those institutions and mechanisms that have always served as deterrents and constraints, are no longer in place. If self policing is no policing, and all organized bodies that are vested with power to police, ultimately become corrupted, what options do we have available to us?

In keeping those dark forces and shadow drivers from overwhelming everything? Will math help, or will itself prove to be some kind of a dark force? Only seeking to be used and ingratiate itself to life, so that our lives grow increasingly misshapen as we need to accommodate it's requirements.

Example of how even the most beneficial thing can turn poisonous

Oh, God, with the swallowing…

Maths Is a Language? Lol. It's an Objective SHIT.

The Ancient Greeks had slaves and pedophilia, therefore, we can learn little from them. …

Very enjoyable lecture, thank you for sharing.

Monk is such an enjoyable lecturer