MathFoundations1: What is a number?

jehovajah Says:

Feb 12, 2011 - 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 Says:

Feb 12, 2011 - 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 Says:

Feb 12, 2011 - Warning, this video is not for university students!However, if you have an open and curious mind, it could be for you!

jehovajah Says:

Feb 12, 2011 - Exactly. And it is very hard for a trained mathematician to do this, but norman is trying and your advice is sound.

jehovajah Says:

Feb 12, 2011 - 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 Says:

Feb 12, 2011 - 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 Says:

Feb 12, 2011 - Final comment i promise, We should start with our apprehension of space, our impulse to measure, and our need to compare and distinguish.

shioswsey Says:

Mar 9, 2011 - 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.

gregg4 Says:

May 11, 2011 - Hi NormanWhat you are talking about looks vaguely like Peano arithmetic. Is that what you are getting at?Gregg

coopclauson Says:

May 16, 2011 - 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 Says:

May 16, 2011 - 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)

njwildberger Says:

Jul 14, 2011 - 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.

plmctellasgr Says:

Jul 20, 2011 - At last ! real 1-1 and onto mapping and proper counting ,the only correct way to face infinity ....

robotadventures Says:

Nov 21, 2011 - wow... i found my channel i will watch a lot. thank you.

GANMath Says:

Dec 20, 2011 - 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.

cmendnba1 Says:

Dec 20, 2011 - 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.

01Simplexity Says:

Dec 21, 2011 - 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 Says:

Dec 21, 2011 - 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.

njwildberger Says:

Dec 22, 2011 - Maybe, maybe not. Such is the way of philosophical bantering, which I have little patience with.

01Simplexity Says:

Dec 23, 2011 - 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!

kingofdice66 Says:

Jan 11, 2012 - 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.

nuqleo Says:

Mar 4, 2012 - interesante

01andak Says:

Apr 10, 2012 - Here begins my journey. Thanks.

JEESherazi Says:

Apr 28, 2012 - I've been arguing for quite some time with my professors about the way students are taught mathematics. We were taught calculus in a way which lacked a lot of rigor and the next year they taught us the correct way. After I asked why didn't they just go straight to the point with the real, logically correct approach, two of my professors said "Students are not ready to understand the real logic of mathematics." What I certainly don't understand is how such approach can work.

facilsempre Says:

May 24, 2012 - So you presented me with natural numbers, a string of ones, and you then told me there is a successor, just put another one to the string, which is a natural number. But then, you told me that we can say which natural is bigger by pairing ones, and the one that has unpaired ones is the bigger one.How do you prove you can pair every number in your string? I mean, how do you say the process will end? You just say so, but there is no proof about it, no axiom to rely on.So, what to do?