WHAT IS A NUMBER? (Real Analysis Series Begins)

A discussion on types of numbers, the Pythagorean Theorem, and Ordered Fields.

[liamschreibman]

Please continue with this series, I would be interested in seeing some more advanced content from your channel.

[OrangePingPongBall]

You do this quality content that is available to us with no cost, thank you Sir!

[sofianesk]

Seeing this title filled me with the same kind of excitement a child feels when his favorite TV show comes up

[thattimestampguy]

1:47 Numbers are Objective Logical Objects, said Gottlob Frege.
3:09 A number is an extension of a concept.
3:41 The number of f’s is equal to the number of g’s if and only if there is a 1 to 1 correspondence.

4:43 Russell’s Paradox.
5:43 Formalist Approach.
6:03 “We must know, we will know.”
6:13 Poincaré’s view.

7:01 Alfred North Whitehead’s Opinion.

*What is a number?*
8:12
_Natural Numbers_
8:35 The Natural Numbers. Aka The Counting Numbers.
_Integers_
8:45 The Integers.
9:34 Integers are whole numbers that can be either positive or negative.

_The Rationals_
9:43 Rational Numbers are NESTED and CLOSED.
10:38 Rational Numbers have problems.

*Problems in The Rationals*
11:04

*The Pythagorean Theorem*
12:13 a^2 + b^2 = c^2
13:07 Pythagoreanism, disproved by Hipposis. Hipposis proved sqrt of 2 is irrational.
14:03 Breaking ground, whether the world was ready for it or not.

*Proving The Sqrt of 2 is Irrational*
14:31
15:06 Assuming p/q is reduced.
16:27 q^2 = 2 * k^2
16:55 The Rationals have almost every other property you want, in an ordered field.

*An Ordered Field*
17:15 a field is a set that satisfies the classic multiplicative and add additive properties.

*the field axioms*
17:40
• associative (a + b) + c) = (a + (b + c)
• communicative
• distributive
• identity
• inverse

18:29
1) The Natural Numbers do not form a field because they failed the law of identity and the law of inverses because the natural numbers do not include negative values.

19:05 2) integer is only failed because they failed the second half of the last axiom the inverses,

20:00 The Rational Numbers form an Ordered Field.

21:15

[Naaroo]

As an ARAB EGYPTIAN i really respect you sir and your content is such a treasure

[SteveS-sk]

For those who like to read, in addition to Frege’s _Die Grundlagen der Arithmetik_, I also recommend his earlier _Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens_ . Also, Richard Dedekind’s _Was sind und was sollen die Zahlen?_ I highly recommend reading these in German if you can; there are some subtleties that get lost in translation. If not, these and other seminal works are translated in _From Frege to Gödel_ by Jean Heijenoort. For a more modern view of the history of the concept of number, try _Geschichte des Zahlbegriffs_ by Helmuth Gericke.

[_Hammy_]

I was scared of this day. I am a calc student but i am excited to try to learn with you!

[noahhayes-tl]

i love watching this dudes videos in the background to my gaming

[coderpro]

one of the best introduction to real analysis,

[Vhaanzeit]

One of the best definitions of 'number' I've found when digging through the history books was in Euler's Elements of Algebra. I can't recite it all here, but the 1828 translation from the French edition of Elements of Algebra provides a really solid. I can summarise it below and quote some parts:

1. 'Whatever is capable of increase of diminution, is called a magnitude or quantity.'
2. There are many types of 'magnitude' in the various fields of Mathematics.
3. 'Mathematics, in general, is the science of quantity; or, the science which investigates the means of measuring quantity.'
4. 'Now, we cannot measure or determine any quantity, except by considering some other quantity of the same kind as known and pointing out their mutual relation.'

There's way more to list in his opening 2 pages, but effectively the definition of number involves firstly setting a baseline object with which to compare (this can be something physical in our universe or something more abstract like setting the standard for a currency) and then comparing other objects of similar types. For example to discuss Voltage, one needs a standard for the Volt in exactly the same way one needs to set a standard with respect to length using the metre/yard. Once we have set the standard for a unit of measure, we can compare other objects of the same type using whole or equal parts of the base unit.

When I looked at Descarte's La Geometrie I and some of Isaac Barrow's Lectures on Geometry, when they name the start and end points of a line segment to be an axis, they have effectively just set the standard unit of measure which which to compare other line segments in their discussions on Geometry. Then you can come to conclusions in geometry about relationships between points and distances.

After geometry, Algebra ends up being the portable 'package' that generalises the concept of setting a unit to be a standard, then enumerating based on that standard. This now allows for a portable/multi-discipline framework to analyse many fields of science as well as more abstract mathematical realms.

Even when we count using the natural numbers, we have to ask 'what are we counting?' If it's in just the world of Algebra, we're count abstract units denoted as 1. If it's the physical world, it could be sheep/ducks/rocks/cards or once again, phenomena like voltage/pressure once a standard has been set. When such a question is asked, we always invoke a standard unit of measure as a result.

I look forward to the Analysis Series. It's a pretty huge topic and I'm here more for the philosophical portions of it! Thanks Prof!

[JR-ucnk]

What a brilliant video! I love the history too! I hope you upload more in this series, it's my favorite so far!

[pratikmaitra]

Truly a great video. There are so many university students(even from STEM background) who do not know anything about fields, groups, rings.

[tiktok-Buumidas]

Thank you for helping me solidify the basics, school does not do this enough!

[gigachadsdad]

Thank you very much for making this! I have been waiting for you to make this very video for a while now and I just want to show my appreciation for what you do. Truly, thank you.

[douglasstrother]

"Dammit, Boomhauer! If you keep violatin' the axioms, I'm gonna kick your ass!" (To be said in "Hank Hill" voice)

[dipankarbanerjee]

Perfect timing sir!! Real Analysis is in my current semester curriculum

[academyofuselessideas]

Pretty neat video. Math education improves greatly with philosophical considerations. It is interesting to see how much opposition different number systems have encountered. Perhaps math is not immune to the Planck's principle: "An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out, and that the growing generation is familiarized with the ideas from the beginning: another instance of the fact that the future lies with the youth."

[KuhuChan]

Thank you for these videos. The clarity is so refreshing, and I love the direct yet mildly humorous style of teaching. Thank you, is all I can say.

[daudkaun]

“If we have to punch your lights out, we will”

our professor is HIM.

[jasoncopin]

I agree math and philosophy go hand and hand. I come to the conclusion that logic is by product of things being true. Your reference to Pythagoras' rejection of irrational numbers makes me think of Darwin. I just his sixth edition of "The Origins of Species" 1872. It is response to Mizart's "On the Genesis of Species" 1871. In the final paragraph of chapter Fourteen, Darwin states that he is so convinced of evolution that he would ignore any fact or argument that does not support evolution. Now this a man who is logical. How much worse is it to deal with people who arrive at an idea illogically? Like in biology and sanitation.