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.

100 thoughts on “Intro to the Philosophy of Mathematics (Ray Monk)

  1. This speech is so shallow and uncapable.. maths itself is not a philosphy but universe(s) it/themselves.. everyhing is not just numbers.. they are just integers.. what existancies are the functions.. just like boxes.. machines processing integers to give results.. and whole maths in fact is odd.. there are actually not sub branches of maths such as geometry or any ather arithmatics etc.. they are just another same things different kind of written things.. you can simply write x2+y2=0 or you can draw it.. they are the same thing.. the remaining is filled with the integers results different conclusions according to the amount.. just multiplies of the same concept same shape..

  2. Abstract art uses a visual language of shape, form, color and line to create a composition which may exist with a degree of independence from visual references in the world.[1] Western art had been, from the Renaissance up to the middle of the 19th century, underpinned by the logic of perspective and an attempt to reproduce an illusion of visible reality. The arts of cultures other than the European had become accessible and showed alternative ways of describing visual experience to the artist. By the end of the 19th century many artists felt a need to create a new kind of art which would encompass the fundamental changes taking place in technology, science and philosophy. The sources from which individual artists drew their theoretical arguments were diverse, and reflected the social and intellectual preoccupations in all areas of Western culture at that time.[2]

  3. "[…a number] doesn't exist in space and time" I heard that a black hole is not matter, it is a pure gravity. What is the relationship, in terms of their form of existence, between black holes and numbers?

  4. I don't think there's any topic deeper than the study of mathematics, God or time.

    This is gold

  5. Definitely having a newfound appreciation for mathematical manipulation and the abstraction necessary for philosophical thought. Thanks for this upload!

  6. Just what I was looking for!
    Great post.
    Here's a mathematical tune called: 0 + 2 = 1 = https://www.youtube.com/watch?v=7Al0_UbBPo8 enjoy I did.

  7. What if platos world of forms are just information which is carried through time by culture/memes? Somebody knows a bit more about this?

  8. why did the problem of irrational numbers raised from the theorem. square root of two has not to do with triangles?

  9. 2+3=5 if it's defined for a flat spacetime geometry, and because all geometries coexist according to the relative time-space duration differential, (ie subject to the methodology of measurement), then it's a (strictly defined) approximation convention worked out in practice from probabilities, …and therefore mathematics is a philosophy of "formative" definitions, adopted in common usage.

    For any mathematics to be "necessarily" true, it must recognize that what is counted/selected as an objective, conforms to the geometrical context, ie the time and position of the apples in question is specifically defined so that the form of objectivity/veracity can be assumed.

    All definite objectives/existence rely on the truth of one second per second being a repeatable measurement. Inherent uncertainty allows the evolution of forms. Because it's the congruence of resonant frequency probability in superposition/measurement against the singularity of certainty, as a focused image, that is the perceived reality. Ie if philosophy is a graduated spectrum of perceived values, then mathematics is the core objective of established social certainties, …to the extent that if a proposed philosophy doesn't have an objective in the mathematical sense, then it's more analytic observation, than the "love of sophistry" or sophistry of love that is the reason for philosophy as a social practice. (?)
    (practical people ask, rhetorically, why bother, if there's no purpose in building, refining and testing a philosophy, mathematical system or science?, knowing that the residual purpose is self-aggrandizement of empty sophistry(?))

  10. I use my advanced knowledge of mathematics with human psychology to predict human behaviour.

  11. Kant was closer to the truth than frege you only need to consult godel todscover that

  12. Its nice to take time post education to have the time to ponder and ruminate on these concepts in Math.

  13. I am surprised to find that I understand some of these Concepts yet I am not mathematically inclined or as they may say have that type of intelligence but philosophy is known as the glue that conjoins all studies in a type of syncretism so maybe that explains it

  14. Regarding his question at 8:00, here are my thoughts: I feel like numbers are about the ability to make distinctions between things. When we say that there are 2 things in front of us, we mean that we can distinction one from the other. There is a certain logic in the distinguishability of things that leads to mathematics of numbers.

    Also the universe could have been one big uniform 'thing', but it isn't – instead you can make distinctions between different parts of the universe and different things in the universe.

  15. I hate math im bad at it whyyyyyy is this on YouTube I mean wtf but I will try I think maybe probably I don't know

  16. like a cut out of a piece of cardboard, math is a man made stencil where you can look through and find pieces of reality.

  17. Interesting… in the Cohen Brothers flick 'A Serious Man', one character purports that, "Mathematics is the art of the possible."

  18. Excellent! Read Ray Monk's biography on Oppenheimer, you can get it through Amazon, now in PB.

  19. Since mathematics is the science of order and logic is the science of making correct inferences then how can they be related?

  20. [Mathematics and Physics – Ideal and Actual Worlds]: Mathematics is often called a language that describes nature or its changes or movements, and is also used as a tool to model logic and judge their right and wrong.

    Is mathematics simply an artificial thing created by human beings? Even its representing symbols are artificial, is a concept or a logic inherent? If it is inherent, does it exist only together with nature, or does it exist more priory and originally than nature?

    The various symbols and expressions used in mathematics must have been defined by humans. But it can not be said that the concepts and theories in it are also artificial things created by humans.

    For example, when describing a natural law with an expression, the law can not be said to have been created by a human being, and when any right logic is expressed by a formula, the logic itself can not be said to have been created by man.

    The law of nature – if the law is right, regardless of whether it is good or bad to human beings – is a description of mathematical modeling of the causal relationship that the same phenomenon always occurs in the same condition in the physical world. It is a logical problem that human beings can not intervene.

    The main task of physicists is to find these laws or principles that exist in nature. And engineers – even though they find such laws or principles themselves – make them available as machines or products mediated through matter, so that the found laws or principles can be used in real life.

    However, the individual cases that occur in nature or are applied in real life correspond to special cases of actual cases that – can be expressed as numerical values – are infinitely possible in these mathematically modeled natural laws and principles. Thus, realization in nature and everyday humans life can be said to be the realization of some cases of mathematics based on matter.

    In this case, if the mathematical logic is wrong, the behavior of the manufactured product will not work consistently with the corresponding principle as it is predictable, and – especially for satellite or missile – remote observing, monitoring, control, adjusting, management and maintenance will not be possible.

    The question is that if there is a perpetual logic that human beings have found or thought out through thinking, even not in nature, that is always judged to be right for anyone who thinks right, transcending the physical space and time, anywhere, anytime, then who made the logic and how can it always remain right?

    Examples of such logic include the consequences of reasoning proven through various pure mathematical axioms or rational thought processes. As an example, the logic of the most basic arithmetic rules (addition, subtraction, multiplication and division) has been found and used separately in different parts of the world in the past when traffic and communication never developed, and the logic that the physical world is finite and always changes has never been denied and confirmed right in natural daily life.

    Thus, mathematics, logic, and philosophy are more primitive than physics, and they are independent of the physical world – or have no relevance to physics – and have been able to study independently. Concept or right reason has been felt to exist before the physical world.

    Plato felt mathematics as the domain of the Idea world and the God’s world. Mathematics deals with infinity, but physics rejects infinity. This is because the world of matter is fundamentally a finite world, and mathematics can handle any logical world away from the physical world.

    For example, the theorem that the sum of two sides is greater than the other side in a triangle – corresponding to the pure concept of logic that the straight line distance is shorter than a certain curve distance – is right in the ideal world, but it is impossible to realize it perfectly in the actual world where refraction occurs even in light.

    Repeating the process of bisecting each line of the oblique sides and attaching them to the base side on an equilateral triangle in an infinite number of turns, results in the final that the sum of the two oblique sides becomes equal to the base, and the above theorem becomes wrong. This means that it is not possible in the actual world to repeat the bisecting without any loss and infinitely.

    The Pythagorean theorem that the square of the base of a right triangle plus the square of the height equals the square of the hypotenuse, and the mathematical formulas that determine the circumference or width of a circle are also ideal cases. In he actual world, we can not make perfectly right triangles or circles without error.

    This is why some industrial products (eg, construction, machinery, electronics, etc.) used in the actual world can not be completely perfect, and – this is because the physical world is ceaselessly and interchangingly changing as a whole – have limited performances and life spans. And we may understand that the fact that the natural exponents and cosmological constants (eg, the ratio of the circumference of a circle to its diameter π, natural exponent e, light speed c, Planck constant h,

    [Difference between mathematics and physics in 0 and infinite handling]:
    Physics deals with the physical world. And mathematics deals with the logic world. Physics deals with a world in which material existence exists, and mathematics deals with a logical world that may not require any material existence.

    Therefore, in physics, zero (0) and infinity are not handled. The physical world treats the change mainly on the assumption that there exist somewhat physically in finite space-time, and it does not handle the case that existence is infinity. In other words, physics deals mainly with the existence of certain physical beings in a finite world.

    However, mathematics can treat zero (0) without physical existence, and also treat infinity. The world without a physical being is a primordial world, and the world of infinity means to contain all possible cases.

    Thus, physics corresponds to special cases of mathematics and can be explained based on mathematics. In other words, the laws of physics appear as mathematical expressions in special cases in various cases in mathematics. This shows that the physical world is a realization of a part of the logical world.

    (Extracted from OnCharm Lee’s book “Humans & Truth”)

  21. Mathematical objects are abstract. What we perceive are their correspondence . It's just like color , that exists in mind only . Different animal see different colours of the same abstract entity. Same is with space and time , means size of the real object is just perception .

  22. Glorious stuff. The speaker evokes many questions within me. This will repay listening to over.

  23. I would say to the question "what is the number?" I would say a number is merely a bit in the informational substrate

  24. Logic as the study of language and ways of putting things together in which some times we can come up with a tautology.

  25. There is Plato and there is Aristotle, the two founding fathers dividing the lot between them. So it went up till Hume who forcefully but elegantly and sweetly tore it down… until Kant very intelligently tried to restore it and thought himself successful. So it went until Wittgenstein tore it all down. After that we are all orphans. Welcome to desperation!

  26. By definition, this is a phenomenally insightful and educational ordered series of acoustic vibrations, synthesizing two subjects of infinite interest.
    Great vid well made

  27. This contains a certain level of bullshit about mathematics being certain and a good basis for philosophy. Everyone who knows anything about science can tell you that it is in physics (and related hard sciences) that one must know what they are talking about, mathematicians do not have this requirement because they do not deal with the real world. And so there are many more physicists (and related) that were philosophers or contributed much to philosophy than mathematicians.

  28. Mathematics is limited, cannot model all of reality. It can't even prove itself ultimately (e.g Godel's incompleteness theorem). Where mathematics ends, physics keep exploring and forcing mathematics to change.

  29. Nothing is contingent. Therefore, it was always going to be black trousers. -Spinoza

  30. Anyone care to explain Russell's proposition that Monk mentioned at the very end about "if there was a god"?

  31. Very good lecture!

    Mathematics is a logical world that always exists right beyond the physical worlds. Mathematics includes all possibilities that the physical world can exist and change spatially. Numbers are associated with possible forms for objects to exist. Thus, mathematics is a study of the relationship between these numbers and the possibility that they can mutually change.

    The physical world is a special case of realization of the world of logic of mathematics through the medium of material. Physics is a study of the physical world that actually exists and changes. The law of physics for changing contains unchanging mathematical truths.

    OnCharm Lee (Author of the book “Humans & Truth – human life is the awakening process”)

  32. You make it sound like synthetic statements are wrong while analitic arent

  33. In the Frege examples of cards, saying "it" has the property of 52 when thinking about cards, and "it" has the property of 4 when thinking of suits seems like an error of reference to me. It implies "It" is the same thing in both cases. They could just as easily be two different collections or objects. There is no "it" having two separate numbers then, but rather "this" and "that", one which is a collection of cards, and the other which is a collection of suits, and each have their objective property of number. In the example, he's just talking about one collection with two numbers, and saying we're thinking about "it" in two different ways, when in fact this can be two different things we are thinking about, each being thought about in ONE way.

  34. 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

  35. Very good, especially for someone trying to make up for 68 years of computer manuals and motorcycle magazines. Thanks.

  36. how doe he justify the statement about Southampton as being on the cost as not necessarily true, i think it was bad example

  37. Iff you wear black trousers it is necessarily true you wear black trousers.

    There's no ways you could wear something else. If that were the case then 2+3=5 is not necessarily true rather it could be 4+4=8 which is necessarily true.

    If you claim anything else you comit false dichotomy fallacy.

  38. If it is unprovable by logic – that means it is unexisting?!!! "Just a myth" mean to make this statement more "logical" perhaps? This is too weak wine.
    Analogy of this ridiculousness will be to try to prove from what sort of matter and and how the CPU/GPU/RAM etc hardware technology – is – using only software lines of codes.
    Good luck with that!

    What is the evident lesson so far – numbers are closed system much like software code. With code you can control the flow of electrons – directly – yet only indirectly. Depending from your perspective. If you are "In the code" or out of It. And the CPU/GPU/RAM etc – are medians – they exist in the two realms simultaneously.

  39. [The truth is never the same as the relative value.]

    The physical world is constantly changing and temporary. The universe in which we live is opened in unity, and thus all the physical entities in our universe are constantly changing and transient.

    The existence of the physical world and all the phenomena that take place in the world are instances in which the reasons are temporarily realized by means of material. Any physical or natural law, which is the causal law for the changes in such a physical world, is not the reason itself.

    But the laws of the physical world about such continual changes contain internally reasons like mathematical logics that never change. We call those reasons realized in our universe as truths.

    As long as we feel and think only of the phenomenon of the physical world, we cannot see any absolute reason or truth but we can only discuss the relative values as each physical entity.

  40. What happens if something is corrected one hundred years after it is setted as "certain"?

    What does it means for mathematics?

    I "think" that mathematics is just a language, and the "real" number properties exists indenpendently from the math language.

    The proof is people that reach math reallities without using the "standar math language". Like Ramanujan. He "saw" mathematics, he didn't "proof" or "deducted" it. He "saw" it.

    The reallity needs to be "observed" and after… you need to find a way to "proof" or "reply" what have you "saw". The "proof" is not "the reallity", it's a tool to share the reallity.

    Like a language… math can be wrong. In evry system of rules you can "forget" one, two or more options… and your system seems to work if anyone don't "find" that options.

    Like pythagoras.

    Math is a fucking and incredible good system. But is not totally certain. Logic is not perfect.

    All human knowledge is based on the human brain… which is not perfect.

    This means is impossible to reach the "Truth"?
    Not really, human brain has the hability to see when is wrong.

  41. Wow! I now understand…"tautology". I have find a third option in an old problem. I call it, in first place "paradox"… in a second time "logic bomb"… but the trueis, "in logic" things can have a "third state": a, b and c.
    (or more than three)
    Denying a does not make b True… because you need to deny c… but nobody knows c exists. Until today.

    Probably the rules of logic are totally ok, but the question is we don't the instrument (our brain) to use them properly without errors.

    Even if you are a great mathematician. Brain love logic fakes in the search of our desires of thruthness.

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

  43. 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

  44. 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?

  45. 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.

  46. 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.

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

  48. 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 🙂

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

  50. 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.

  51. 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

  52. 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'

  53. Very informative lecture and given with great examples to consider

  54. This is beautiful. I cannot believe he managed to sum up all of that into 32 minutes.

  55. using mathematical construct and using mathematical preponderance of ranking system

    mathematics will always be lower ranking than reality

    mathematics will always be lower ranking than imagination

    There for mathematics will never ever be able to explain the universe or reality

  56. This is one of the most brilliant and interesting lectures I’ve ever listened to. It’s fascinating. Thanks for posting.

  57. I find it funny that YouTube commenters are so smart they will quickly say that science and scientists and general smart people don't believe in God, and that science and the laws of nature and physics disprove God, and then overlook the fact that 67.5% of Nobel Prize laureates in the natural sciences are literally Bible-believing Christians despite being ground-breaking geniuses in things like math, physics, and on.

    Lol at the moody atheists. Atheism has no more substance than saying, "I itch."

  58. Archons and Reptilians love math and money — they get so fat into it and cluster their hive mind networks together to serve themselves and take from everyone else they forget about any compassion or truth for the masses along the way.

    Central banks are a physical reflection of this

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