The "proof by example" of something like n*m=m*n does not seem much more satisfactory than simply saying: try for any two numbers n and m to calculate each value, and observe whether they are equal (and if they are not, then look for the mistake in your calculation, because they are supposed to be). Sure, for any two numbers n and m you can present a separate proof. The interesting thing about Peano arithmetic is that you have induction and one proof that works for all n and m.
And a last comment: Peano's axioms are strong enough to prove properties of addition and multiplication like commutativity, associativity, etc., so again, I think it's more accurate to characterize these as defining the natural numbers, and not the axiom of infinity.
Thanks for your comments. First of all, and most importantly, I object strongly to the use of "Axioms" in mathematics. The current use of these is to introduce unwarranted assumptions into the subject, for which there is no need. Clear definitions are what is needed, not assumptions. As for von Neumann's "definition" of the natural numbers---it is completely unnatural and awkward.
Just to further reinforce the point, I think in the original (non-von Neuman) form, the sequence given in the axiom of infinity looked something like this:
0 = {}
1 = {0} = {{}}
2 = {1} = {{{}}}
3 = {2} = {{{{}}}}
...
The point of this is that the actual construction of the set is less important than the fact that it exists and that it obeys Peano's axioms--in fact, the unary notation you present could simply be interpreted as another model for Peano's axioms.
The really critical question is: does there exist an ''infinite set" of ''all natural numbers''. To even make sense of this, one FIRST needs to define what is meant by an "infinite set" and then prove that one can construct one (say by using all the natural numbers). This program fails because up to now (2011) no-one has been able to perform the first task in a convincing and clear way. All current definitions are essentially mumbo-jumbo.
@njwildberger This is something that I have a bit of difficulty with, because I use the concepts of infinite sets fairly regularly in computer science.
The ones that arise most for me are recursively enumerable ones. These are infinite sets that consist of a description of a finite number of elements that are in the set, and a finite procedure for constructing new elements in the set given existing ones. I think these might meet your criteria because they have finite descriptions.
With that out of the way, the main thing that I'm going to question is the presentation of this set-based construction as defining the natural numbers. In most contexts that I'm aware of, the actual definition is done by Peano's axioms. The only purpose of the axiom of infinity is to assert the existence of a model for these axioms, and also some people like it because you no longer have to talk about mathematical objects that aren't sets.
While I think these videos are interesting and some of the statements provocative, I'm going to object to some of it as I think that what's being attacked is being mischaracterized.
First: in another article, Prof. Wildeberger characterized the axiom of infinity as stating "there exists an infinite set". This is just a digest version--the Wikipedia page for axiom of infinity contains the full statement, which contains the set-based construction talked about here.
...¿No parece haber un misterio en esta posibilidad de reconocer algo como uno, es decir, su unidad? ¿no parece ya haber un mundo completo de cuestiones supuestas en el simple y más básico paso de pasar de la pizarra en blanco al signo que representa el número 1? ¿No parece estar la matemática, en sus bases más simples y primeras, estar estrechamente relacionada con especulaciones no matemáticas?
Cuando contamos contamos unos (ones) y esto se nos da en la experiencia como algo básico y obvio: vemos "algo" y lo identificamos como "algo-uno". Identificamos, por lo tanto una unidad en la cosa contada. Si esto es así, ¿no habría que decir que la posibilidad de toda matemática está en el hecho de poder identificar la unidad de la cosa contada? Vemos un caballo y no un cúmulo indeterminado de distintas sensaciones borrosas. ¿No parece haber un misterio en esta posibilidad de reconocer algo...
Me gustaría hacerte una pregunta. Es interesante el primer paso que muestras en el primer video, aquel de la pizarra blanca al primer signo (which represent the number one). Según esto, parece desprenderse de ahí que los números naturales en realidad no son más que distintas acumulaciones de unos (ones), o, lo que parece ser lo mismo, que el único número que realmente "existe" es el uno, pues todo se resuelve finalmente en el número 1 ... continúo
While the approach with unary system is highly commendable, the "proofs" with concrete numbers in this video border on cheating. They are actually only examples and require the nontrivial atomic judgment if something is "like that" in general. Students might jump to premature conclusions later on in mathematics. I would say graphical examples with rectangular boxes for associativity and two rectangles for distributivity are more appropriate general proofs.
GREAT videos! Who is the intended audience of these foundations of math videos? At my school, Foundations of Math is the class math majors take after Calculus II, designed to prepare them for Abstract Algebra and Analysis. As a math major, I found these videos very helpful. For someone just starting to learn math, a more computational approach to the foundations of math would probably be more appropriate. But for a math major about to take Abstract Algebra, these videos are AWESOME
They are meant for anyone with an interest in mathematics. While mostly elementary, I will be saying things of interest also to students of mathematics and mathematicians.
Mathematics majors ought to watch all these videos very carefully.
Mr. Wildberger, I cannot start to thank you enough for your math foundation video uploads. Am beginning to finally make sense of math, rather than rely of guess work. Really appreciate it.
The intuitive proof of the commutative law asks us to think of "n x m" as a rectangle that can be flipped without affecting its area. This idea obviously works for natural numbers and might (?) be extended to fractions and negatives. But it seems to me that it won't work where "n" and "m" are irrational, because there the sides of the rectangle can't be divided into a finite number of equal parts. Therefore it would be inadequate to think of that rectangle as consisting of rows and columns.
@langengro If you can extend it to fractions, you can extend it to irrationals through induction. If you can prove something about a ratio a/b, and you have an infinite series of a's and b's such that a/b converges to an irrational number c, you've proved that c has that property, too.
since we are interested in counts, rather than things, the things being counted can be replaced with other things. The things can be selected to suit different needs. Vertical marks are suitable for counting on a fixed horizontal line. Circles are suitable for counting on a rotating surface. Rotating a count doesn't affect the counting. Two counts that can be seen as identical after a change that doesn't affect the counting are identical. Operations that have identical results are equal.
haha flipping just look like transposed matrices
00Gravity 4 months ago
The "proof by example" of something like n*m=m*n does not seem much more satisfactory than simply saying: try for any two numbers n and m to calculate each value, and observe whether they are equal (and if they are not, then look for the mistake in your calculation, because they are supposed to be). Sure, for any two numbers n and m you can present a separate proof. The interesting thing about Peano arithmetic is that you have induction and one proof that works for all n and m.
tommyrjensen 8 months ago
And a last comment: Peano's axioms are strong enough to prove properties of addition and multiplication like commutativity, associativity, etc., so again, I think it's more accurate to characterize these as defining the natural numbers, and not the axiom of infinity.
coopclauson 10 months ago
Hi coopclauson
Thanks for your comments. First of all, and most importantly, I object strongly to the use of "Axioms" in mathematics. The current use of these is to introduce unwarranted assumptions into the subject, for which there is no need. Clear definitions are what is needed, not assumptions. As for von Neumann's "definition" of the natural numbers---it is completely unnatural and awkward.
njwildberger 10 months ago
Just to further reinforce the point, I think in the original (non-von Neuman) form, the sequence given in the axiom of infinity looked something like this:
0 = {}
1 = {0} = {{}}
2 = {1} = {{{}}}
3 = {2} = {{{{}}}}
...
The point of this is that the actual construction of the set is less important than the fact that it exists and that it obeys Peano's axioms--in fact, the unary notation you present could simply be interpreted as another model for Peano's axioms.
(continued)
coopclauson 10 months ago
Hi coopclauson,
The really critical question is: does there exist an ''infinite set" of ''all natural numbers''. To even make sense of this, one FIRST needs to define what is meant by an "infinite set" and then prove that one can construct one (say by using all the natural numbers). This program fails because up to now (2011) no-one has been able to perform the first task in a convincing and clear way. All current definitions are essentially mumbo-jumbo.
njwildberger 10 months ago
@njwildberger This is something that I have a bit of difficulty with, because I use the concepts of infinite sets fairly regularly in computer science.
The ones that arise most for me are recursively enumerable ones. These are infinite sets that consist of a description of a finite number of elements that are in the set, and a finite procedure for constructing new elements in the set given existing ones. I think these might meet your criteria because they have finite descriptions.
coopclauson 10 months ago
With that out of the way, the main thing that I'm going to question is the presentation of this set-based construction as defining the natural numbers. In most contexts that I'm aware of, the actual definition is done by Peano's axioms. The only purpose of the axiom of infinity is to assert the existence of a model for these axioms, and also some people like it because you no longer have to talk about mathematical objects that aren't sets.
(continued)
coopclauson 10 months ago
While I think these videos are interesting and some of the statements provocative, I'm going to object to some of it as I think that what's being attacked is being mischaracterized.
First: in another article, Prof. Wildeberger characterized the axiom of infinity as stating "there exists an infinite set". This is just a digest version--the Wikipedia page for axiom of infinity contains the full statement, which contains the set-based construction talked about here.
(continued)
coopclauson 10 months ago
...¿No parece haber un misterio en esta posibilidad de reconocer algo como uno, es decir, su unidad? ¿no parece ya haber un mundo completo de cuestiones supuestas en el simple y más básico paso de pasar de la pizarra en blanco al signo que representa el número 1? ¿No parece estar la matemática, en sus bases más simples y primeras, estar estrechamente relacionada con especulaciones no matemáticas?
Gracias.
D.
diegocolomes 1 year ago
Cuando contamos contamos unos (ones) y esto se nos da en la experiencia como algo básico y obvio: vemos "algo" y lo identificamos como "algo-uno". Identificamos, por lo tanto una unidad en la cosa contada. Si esto es así, ¿no habría que decir que la posibilidad de toda matemática está en el hecho de poder identificar la unidad de la cosa contada? Vemos un caballo y no un cúmulo indeterminado de distintas sensaciones borrosas. ¿No parece haber un misterio en esta posibilidad de reconocer algo...
diegocolomes 1 year ago
Muy buenos videos Norman. Muy explicativos!
Me gustaría hacerte una pregunta. Es interesante el primer paso que muestras en el primer video, aquel de la pizarra blanca al primer signo (which represent the number one). Según esto, parece desprenderse de ahí que los números naturales en realidad no son más que distintas acumulaciones de unos (ones), o, lo que parece ser lo mismo, que el único número que realmente "existe" es el uno, pues todo se resuelve finalmente en el número 1 ... continúo
diegocolomes 1 year ago
While the approach with unary system is highly commendable, the "proofs" with concrete numbers in this video border on cheating. They are actually only examples and require the nontrivial atomic judgment if something is "like that" in general. Students might jump to premature conclusions later on in mathematics. I would say graphical examples with rectangular boxes for associativity and two rectangles for distributivity are more appropriate general proofs.
vlddas 1 year ago
You should write a book on this, you can make some major money. You have a gift for making things less complex.
KillSwitchOnSteriods 1 year ago
GREAT videos! Who is the intended audience of these foundations of math videos? At my school, Foundations of Math is the class math majors take after Calculus II, designed to prepare them for Abstract Algebra and Analysis. As a math major, I found these videos very helpful. For someone just starting to learn math, a more computational approach to the foundations of math would probably be more appropriate. But for a math major about to take Abstract Algebra, these videos are AWESOME
tyler121515 2 years ago
They are meant for anyone with an interest in mathematics. While mostly elementary, I will be saying things of interest also to students of mathematics and mathematicians.
Mathematics majors ought to watch all these videos very carefully.
njwildberger 2 years ago
Comment removed
Ormaaj 2 years ago
Comment removed
Ormaaj 2 years ago
nice shirt
jwh335 2 years ago
Mr. Wildberger, I cannot start to thank you enough for your math foundation video uploads. Am beginning to finally make sense of math, rather than rely of guess work. Really appreciate it.
VivekFerr 2 years ago
Hi,
I am glad you are getting something out of them. Stick with it!
Norman
njwildberger 2 years ago
The intuitive proof of the commutative law asks us to think of "n x m" as a rectangle that can be flipped without affecting its area. This idea obviously works for natural numbers and might (?) be extended to fractions and negatives. But it seems to me that it won't work where "n" and "m" are irrational, because there the sides of the rectangle can't be divided into a finite number of equal parts. Therefore it would be inadequate to think of that rectangle as consisting of rows and columns.
langengro 2 years ago
@langengro If you can extend it to fractions, you can extend it to irrationals through induction. If you can prove something about a ratio a/b, and you have an infinite series of a's and b's such that a/b converges to an irrational number c, you've proved that c has that property, too.
sco08y 1 year ago
Thank you. Please let other people know about this resource.
njwildberger 2 years ago
Thank you for explaining mathematics in such an intuitive and basic manner.
aroguegurklis 2 years ago
I'm glad you find the video(s) useful. If you have any questions, let me know through these comments, and I will try to answer/explain.
njwildberger 2 years ago
since we are interested in counts, rather than things, the things being counted can be replaced with other things. The things can be selected to suit different needs. Vertical marks are suitable for counting on a fixed horizontal line. Circles are suitable for counting on a rotating surface. Rotating a count doesn't affect the counting. Two counts that can be seen as identical after a change that doesn't affect the counting are identical. Operations that have identical results are equal.
otherchaz 3 years ago