Thank you for all the work you are putting into making these videos, it is much appreciated by me and others. Keep up the good work cause this is pure GOLD.
3. On your rejection of infinite sets, for example, if objects have no existence independent of notations, isn't it sufficient to define an infinite set symbolically for it to exist? The obvious counterargument to this is that even if this were done, then this definition would be inadequate to reason about infinite sets, but actually the axiom of infinity does provide a construction similar to the one you do here (set of natural numbers contains a number, and for each number its successor)
I'm a computer engineering student, and am interested in the foundations of mathematics, and am finding these to be interesting. A few questions that come up:
1. This seems to me to be a revival of the intuitionist school of thought, like Kronecker, etc. Do you consider this to be what you're doing?
2. One feature of your system that I'm noticing is that mathematical objects don't have existence independent of the notation used to represent them. Would you agree with this?
I do not subscribe to any School of thought. The question of whether mathematical objects do or don't have an existence seems somewhat philosophical to me, i.e. largely a waste of time.
We write down certain expressions, manipulate them in certain ways, and interpret the results in various ways. This is our simple-minded human way to explore the mathematical world, whose true reality is unfathomable to us.
since you distinguish between our simple-minded way of understanding mathematics and "true" mathematical reality, it would seem that you are already a Platonist. As a result, you are taking a philosophical position -- even if you find it to be a waste of time.
@njwildberger You are, of course, entitled to your feelings. I was merely making the amusing point that you cannot escape philosophical assumptions. Indeed an anti-philosophical position is just a philosophical position of another kind -- and a self-undermining one at that!
My reply, is a riddle: so are you now making white black, or space empty? Fundamentally this is what we have done at the foundation of mathematics in the modern era, something that even the greeks and Indians did not and could not do.
Hoping you are being provoked to think, not riled by any unintended other interpretation.
Honestly, this is meat and potatoes to me. its great stuff.
A historical survey is usually how we start, but our historical assumptions are often undermined by archaeological evidence. So both a historical, archaeological, and human developmental biological approach i have found to be the most instructive and revealing.
After all something wrong must be going on if you can call something "nothing" don't you think!
To be plain, you are standing in front of a white board and telling me that "this is nothing".
Ok , you start here as i woul expect any mathematician trying to dig the foundations would, and how i started. But to start on the wrong premise is to make the whole exercise futile.
This will engage with most mathematically trained and constrained mindsets, but, and this is a serious provoking question: would it engage with an aboriginal mindset?
Hey buddy, you did not misunderstand. Somebody is trying to divert you. Do not get hung up on categories, follow your nose or your gut instinct. Mathematics is at stake here!
This is not a criticism, for this work is absolutely marvelous. However, math breaches the rules a bit when mathematicians define the number one. For while "one" can fairly be applied to something abstract, when we apply it to tangible things, we have a problem. That is, if I have ten apple seeds, and say that each one taken by itself is one, and yet they are composed of differing quantities of atoms, then they are in reality much different from each other quantitatively.
I wrote it in a very philosophical way for the following reason. I want to see the border line, out of which mathematics would not be so strong and legitimate, I want to see where it falls to see what is it stands on.
It's very interesting (at least for me) to go from nothing. But what if we go from really nothing? What is an object? What gives us right to count them? Why we should rely on our fillings that something is correct?
I think that numbers are existing only in our parallel (mind) world, created by us on the basis of what we see, feel, hear etc. This world may have nothing similar with the reality. We grasp only some parts of the reality (do we?), and numbers came from "there"...
Hi. I watched the video and I didn't get any smarter! You said in the beginning "...in a new and better way...", but I feel deceived. You should put up the warning that the video is not for university students.
this is basically a really long way to go about understanding algebra (as in groups, fields etc). Your methods are fine and the student may learn more fully, but will take much longer to learn than the 'normal' way, in my opinion. This is a much more idealistic than realistic method of teaching/learning
So you claim to understand the foundations because it is all algebra? So what is algebra?
This is one among many of the biggest disconnects from mathematics that i find among students.
Although you could give an answer, it most assuredly does not settle the matter. My question is rhetorical because a debate about categories and categorisation is not required here.
From the beginning you assume the axiom of infinity for sets. You have to work somewhere; otherwise you are not doing anything. Nobody claims that sets exist in nature. We simply choose to work in a system where we assume set theory to be consistent. Mathematicians tried to do what you are doing for hundreds of years. You can't make it work in less than ten minutes. In fact, rigourizing this discussion takes us exactly where you claim we don't have to go.
Hi cmendnba1, Your use of the term ``you'' is unclear. If you mean me, then you are mistaken, in that I certainly do not use infinite sets. How could I? They don't extist.
I don't intend to make all this work in ten minutes. This is the first of a rather extensive series of lectures. Keep viewing, then decide whether my formulation makes sense.
@njwildberger I think it is a little unfair for someone to just claim that infinite sets do not exist. I don't think any mathematician would claim that they do in "nature." But that's not what mathematics is about. If anyone ever shows that ZFC is consistent (which I think has been shown to be impossible), would you still have a problem using it as a meta-language in mathematics?
Hi cmendnba1, Whether it is fair or not is surely not as important as whether it is correct or not. As the series progresses you will find plenty of evidence to support my claim. As for ZFC, to me that framework is seriously flawed, because the meanings of the words and concepts used are not sufficiently precise---as I will explain later in this series.
Can you explain the foundations of mathematics to an interested ten year old? If not, you are on the wrong track.
Sorry for two posts at once, but I couldn't resist! I think the foundations of mathematics -- as viewed from ZFC or NFU or any other major system -- can be explained to an interested ten year old. He will not, of course, understand the details, but the general project is quite intelligible. It is the general idea that matters; details are not so significant and are easy to get lost in.
@cmendnba1 the issue is not whether infinite sets do not exist, but even whether infinity exists at all. philosophically, it is a sham; how can you quantify something that never ends? mathematically, it is still a debate. i am on norman's side here, basing my argument on the philosophical aspect.
@GANMath Well no, I doubt anyone will ever observe an object called infinity. I think that's silly. For me at least, the only thing that matters in mathematics and philosophy is how one reasons. The words "finite" and "infinite" are just adjectives used to describe observations. Saying that the set of natural numbers is infinite is just a compact and easily understood way of saying that given any finite collection of objects, there is no way to pair the objects with all every natural number.
"A natural number is a string of ones." Is zero then a natural number? I have heard that it is and that it isn't, and this definition is confusing for me.
In the system I propose, zero is not a natural number. I realize there are different conventions around, this one seems to me more aligned with the historical approach, in which the concept of zero as a number occurs much later, with the Hindus.
@njwildberger Thank you for clarifying that, but wouldn't it still be good to teach my children about zero when I teach them natural numbers? No apples is a simple concept to understand and it will probably make it easier for them in the long run.
@thekkl Of course zero is an important number, but it is an integer, not a natural number. It does not have the immediacy of the numbers one, two and three..
Keep watching the videos, and you will see we will introduce integers shortly.
I like this video, but I think your "pairing and comparing" idea is based on the skill of making relations. I may be biased toward the current view of mathematics, but I see two set of marks being related to another. This reminds me of set theory, catergory theory and even abstract algebra. I think of one the mathematical axioms we math minds cannot escape is that mathematics is based on the relation.
You touch the marker to the white surface, make a mark, and we agree me that that mark is 'one.'
Then the idea of successor. You make another mark. not on the same place on the surface, using ink from further up in the marker, and we agree that the new mark is related to the first.
Without that agreement, the math halts.
Even if If I count alone both roles are there, I hold the marker, and behold the result.
I'm really interested in your critisism of set theory and analysis, though. If I'm right, you seem to advocate some kind of finitism? Then, I'm looking forward to your views on induction, real numbers and physics. I hope this series will cover those topics.
Secondly, isn't what you are doing just the way numbers were thought about until the 19th century? Nothing new about that...
So, I think this series will be a great resource for teachers and other people interested in elementary mathematics, and does really well in establishing that you really don't need set theory for that. But this "create a new framework for mathematic's foundations" seems a bit far-fetched. After all, you would still need set theory in higher math, e.g. for measure theory.
"A number is a string of strokes on paper", well, isn't that a bit, well, underwhelming? I would say something like "a number is a property of sets of things; namely, the property which any two sets have in common, that are 'equal in number'. This means, there elements can be paired up." Then, you can simply say "we REPRESENT numbers by strings of strokes on paper".
interesante
nuqleo 2 days ago
Thank you for all the work you are putting into making these videos, it is much appreciated by me and others. Keep up the good work cause this is pure GOLD.
kingofdice66 1 month ago
wow... i found my channel i will watch a lot. thank you.
robotadventures 3 months ago
At last ! real 1-1 and onto mapping and proper counting ,the only correct way to face infinity ....
plmctellasgr 7 months ago
Would you recommend Russell's Principia Mathematica to get a good detailed understanding of mathematics as a whole?
atypicalguy 7 months ago
3. On your rejection of infinite sets, for example, if objects have no existence independent of notations, isn't it sufficient to define an infinite set symbolically for it to exist? The obvious counterargument to this is that even if this were done, then this definition would be inadequate to reason about infinite sets, but actually the axiom of infinity does provide a construction similar to the one you do here (set of natural numbers contains a number, and for each number its successor)
coopclauson 9 months ago
I'm a computer engineering student, and am interested in the foundations of mathematics, and am finding these to be interesting. A few questions that come up:
1. This seems to me to be a revival of the intuitionist school of thought, like Kronecker, etc. Do you consider this to be what you're doing?
2. One feature of your system that I'm noticing is that mathematical objects don't have existence independent of the notation used to represent them. Would you agree with this?
(continued)
coopclauson 9 months ago
Hi coopclauson,
I do not subscribe to any School of thought. The question of whether mathematical objects do or don't have an existence seems somewhat philosophical to me, i.e. largely a waste of time.
We write down certain expressions, manipulate them in certain ways, and interpret the results in various ways. This is our simple-minded human way to explore the mathematical world, whose true reality is unfathomable to us.
njwildberger 7 months ago
@njwildberger
since you distinguish between our simple-minded way of understanding mathematics and "true" mathematical reality, it would seem that you are already a Platonist. As a result, you are taking a philosophical position -- even if you find it to be a waste of time.
01Simplexity 2 months ago in playlist MathFoundations
@01Simplexity Maybe, maybe not. Such is the way of philosophical bantering, which I have little patience with.
njwildberger 2 months ago
@njwildberger You are, of course, entitled to your feelings. I was merely making the amusing point that you cannot escape philosophical assumptions. Indeed an anti-philosophical position is just a philosophical position of another kind -- and a self-undermining one at that!
01Simplexity 2 months ago
Hi Norman
What you are talking about looks vaguely like Peano arithmetic. Is that what you are getting at?
Gregg
gregg4 10 months ago
Now every time I see a serious looking old guy I'm going to wonder if he secretly has littlest pet shop pets in his pockets.
shioswsey 1 year ago
Final comment i promise, We should start with our apprehension of space, our impulse to measure, and our need to compare and distinguish.
jehovajah 1 year ago
My reply, is a riddle: so are you now making white black, or space empty? Fundamentally this is what we have done at the foundation of mathematics in the modern era, something that even the greeks and Indians did not and could not do.
Hoping you are being provoked to think, not riled by any unintended other interpretation.
Honestly, this is meat and potatoes to me. its great stuff.
jehovajah 1 year ago
A historical survey is usually how we start, but our historical assumptions are often undermined by archaeological evidence. So both a historical, archaeological, and human developmental biological approach i have found to be the most instructive and revealing.
After all something wrong must be going on if you can call something "nothing" don't you think!
To be plain, you are standing in front of a white board and telling me that "this is nothing".
jehovajah 1 year ago
Ok , you start here as i woul expect any mathematician trying to dig the foundations would, and how i started. But to start on the wrong premise is to make the whole exercise futile.
This will engage with most mathematically trained and constrained mindsets, but, and this is a serious provoking question: would it engage with an aboriginal mindset?
jehovajah 1 year ago
Oh, recently I understood what I missed to understand :). My question refers to metamathematics rather than simple mathematics...
longhorn4500 1 year ago
@longhorn4500
Hey buddy, you did not misunderstand. Somebody is trying to divert you. Do not get hung up on categories, follow your nose or your gut instinct. Mathematics is at stake here!
jehovajah 1 year ago
This is not a criticism, for this work is absolutely marvelous. However, math breaches the rules a bit when mathematicians define the number one. For while "one" can fairly be applied to something abstract, when we apply it to tangible things, we have a problem. That is, if I have ten apple seeds, and say that each one taken by itself is one, and yet they are composed of differing quantities of atoms, then they are in reality much different from each other quantitatively.
sirwizardoflight 1 year ago
@sirwizardoflight
Excellent! So for that reason the Greeks did not count one as an arithmoi. It is the foundation of any scalar scheme we want to set up!
jehovajah 1 year ago
I wrote it in a very philosophical way for the following reason. I want to see the border line, out of which mathematics would not be so strong and legitimate, I want to see where it falls to see what is it stands on.
longhorn4500 1 year ago
@longhorn4500
Exactly. And it is very hard for a trained mathematician to do this, but norman is trying and your advice is sound.
jehovajah 1 year ago
It's very interesting (at least for me) to go from nothing. But what if we go from really nothing? What is an object? What gives us right to count them? Why we should rely on our fillings that something is correct?
I think that numbers are existing only in our parallel (mind) world, created by us on the basis of what we see, feel, hear etc. This world may have nothing similar with the reality. We grasp only some parts of the reality (do we?), and numbers came from "there"...
longhorn4500 1 year ago
Hi. I watched the video and I didn't get any smarter! You said in the beginning "...in a new and better way...", but I feel deceived. You should put up the warning that the video is not for university students.
roger7c297 1 year ago
@roger7c297
Warning, this video is not for university students!
However, if you have an open and curious mind, it could be for you!
jehovajah 1 year ago
this is basically a really long way to go about understanding algebra (as in groups, fields etc). Your methods are fine and the student may learn more fully, but will take much longer to learn than the 'normal' way, in my opinion. This is a much more idealistic than realistic method of teaching/learning
jewbinson 1 year ago
@jewbinson
So you claim to understand the foundations because it is all algebra? So what is algebra?
This is one among many of the biggest disconnects from mathematics that i find among students.
Although you could give an answer, it most assuredly does not settle the matter. My question is rhetorical because a debate about categories and categorisation is not required here.
jehovajah 1 year ago
From the beginning you assume the axiom of infinity for sets. You have to work somewhere; otherwise you are not doing anything. Nobody claims that sets exist in nature. We simply choose to work in a system where we assume set theory to be consistent. Mathematicians tried to do what you are doing for hundreds of years. You can't make it work in less than ten minutes. In fact, rigourizing this discussion takes us exactly where you claim we don't have to go.
cmendnba1 1 year ago
Hi cmendnba1, Your use of the term ``you'' is unclear. If you mean me, then you are mistaken, in that I certainly do not use infinite sets. How could I? They don't extist.
I don't intend to make all this work in ten minutes. This is the first of a rather extensive series of lectures. Keep viewing, then decide whether my formulation makes sense.
njwildberger 1 year ago
@njwildberger I think it is a little unfair for someone to just claim that infinite sets do not exist. I don't think any mathematician would claim that they do in "nature." But that's not what mathematics is about. If anyone ever shows that ZFC is consistent (which I think has been shown to be impossible), would you still have a problem using it as a meta-language in mathematics?
cmendnba1 1 year ago
Hi cmendnba1, Whether it is fair or not is surely not as important as whether it is correct or not. As the series progresses you will find plenty of evidence to support my claim. As for ZFC, to me that framework is seriously flawed, because the meanings of the words and concepts used are not sufficiently precise---as I will explain later in this series.
Can you explain the foundations of mathematics to an interested ten year old? If not, you are on the wrong track.
njwildberger 1 year ago 3
@njwildberger
Sorry for two posts at once, but I couldn't resist! I think the foundations of mathematics -- as viewed from ZFC or NFU or any other major system -- can be explained to an interested ten year old. He will not, of course, understand the details, but the general project is quite intelligible. It is the general idea that matters; details are not so significant and are easy to get lost in.
01Simplexity 2 months ago in playlist MathFoundations
@cmendnba1 the issue is not whether infinite sets do not exist, but even whether infinity exists at all. philosophically, it is a sham; how can you quantify something that never ends? mathematically, it is still a debate. i am on norman's side here, basing my argument on the philosophical aspect.
GANMath 2 months ago in playlist MathFoundations
@GANMath Well no, I doubt anyone will ever observe an object called infinity. I think that's silly. For me at least, the only thing that matters in mathematics and philosophy is how one reasons. The words "finite" and "infinite" are just adjectives used to describe observations. Saying that the set of natural numbers is infinite is just a compact and easily understood way of saying that given any finite collection of objects, there is no way to pair the objects with all every natural number.
cmendnba1 2 months ago
Wildberger, you're making a bloody real contribution to mathematics education. For God's sake, keep going with it. Well done, chap.
peterhi503 1 year ago
but you're as an axiom that the number one exists
affablegiraffable 1 year ago
"A natural number is a string of ones." Is zero then a natural number? I have heard that it is and that it isn't, and this definition is confusing for me.
thekkl 1 year ago
Hi thekkl
In the system I propose, zero is not a natural number. I realize there are different conventions around, this one seems to me more aligned with the historical approach, in which the concept of zero as a number occurs much later, with the Hindus.
njwildberger 1 year ago
@njwildberger Thank you for clarifying that, but wouldn't it still be good to teach my children about zero when I teach them natural numbers? No apples is a simple concept to understand and it will probably make it easier for them in the long run.
thekkl 1 year ago
@thekkl Of course zero is an important number, but it is an integer, not a natural number. It does not have the immediacy of the numbers one, two and three..
Keep watching the videos, and you will see we will introduce integers shortly.
njwildberger 1 year ago
Great videos! Excelent teaching! Plan to watch all of them. Thank you.
Retired but still student; mostly math.
nandoorihuelsimon 2 years ago
The beginning reminds me of George Spencer Brown.
langengro 2 years ago
really awesome. thank you for your offer and contributions!
lisp21 2 years ago
the starting point is 3:42
iwantcoolname 2 years ago
42... than number again...
iwantcoolname 2 years ago
I wish you were my math teacher when I was youn
riponKS89 2 years ago
Professor, these videos have really opened my mind, thank you so much!
riponKS89 2 years ago
This was an amazing video and I am looking forward to seeing many more.Dr.Wildberger explains math in a new and innovative way...Well done!!!!
Margot1959 2 years ago
I like this video, but I think your "pairing and comparing" idea is based on the skill of making relations. I may be biased toward the current view of mathematics, but I see two set of marks being related to another. This reminds me of set theory, catergory theory and even abstract algebra. I think of one the mathematical axioms we math minds cannot escape is that mathematics is based on the relation.
mathgeek37 3 years ago
Counting is a conversation.
You touch the marker to the white surface, make a mark, and we agree me that that mark is 'one.'
Then the idea of successor. You make another mark. not on the same place on the surface, using ink from further up in the marker, and we agree that the new mark is related to the first.
Without that agreement, the math halts.
Even if If I count alone both roles are there, I hold the marker, and behold the result.
otherchaz 3 years ago
I'm really interested in your critisism of set theory and analysis, though. If I'm right, you seem to advocate some kind of finitism? Then, I'm looking forward to your views on induction, real numbers and physics. I hope this series will cover those topics.
Pebbe496 3 years ago
Secondly, isn't what you are doing just the way numbers were thought about until the 19th century? Nothing new about that...
So, I think this series will be a great resource for teachers and other people interested in elementary mathematics, and does really well in establishing that you really don't need set theory for that. But this "create a new framework for mathematic's foundations" seems a bit far-fetched. After all, you would still need set theory in higher math, e.g. for measure theory.
Pebbe496 3 years ago
I really like your videos. Great work!
But a few comments about this new series.
"A number is a string of strokes on paper", well, isn't that a bit, well, underwhelming? I would say something like "a number is a property of sets of things; namely, the property which any two sets have in common, that are 'equal in number'. This means, there elements can be paired up." Then, you can simply say "we REPRESENT numbers by strings of strokes on paper".
I think this explanation is truer to the notion.
Pebbe496 3 years ago
The journey begins :)
CPLains 3 years ago