Category Theory: Exploring Math’s Enigmas With Eugenia Cheng (362)

Is math real? How to bake pi? And how much is x+y, really?

Many people don't like math because they find it too complicated or boring. But math can actually be a lot of fun, and we can find it everywhere in life, even in the most mundane things like baking. And it is through baking that today's guest, Eugenia Cheng, decided to rid the world of math phobia.

Dr. Cheng is a renowned mathematician, educator, author, and concert pianist. She's a scientist in residence at the School of the Art Institute of Chicago, where she teaches mathematics to art students. She is an expert in category theory and has recently published a book, Is Math Real?, which we will discuss in detail today!

Join Eugenia and me as we explore mathematics’ deepest truths.

Join this channel to get access to perks:

Intro (00:00)
Judging a book by its cover: Is Math Real? (00:54)
On the unreasonable power of mathematics in the physical sciences (03:50)
If there were no physical world, would math exist? (07:58)
The number zero (10:14)
Is our brain a massive computer? (16:46)
How to Bake Pi (21:52)
Category theory (26:21)
How to revitalize and modernize education (38:36)
Is math created or discovered? (44:26)
Outro (48:47)

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[radientbeing]

The total number(cardinal number) of different mathematical geometric constructions (smooth/non-smooth) is equal to 2(mathematical point, missing point)^aleph1 = aleph2; this is equal for 1-dimensional, 2-dimensional, 3-dimensional spaces etc. I lean towards discovering rather than inventing this.

[guidor.]

I believe "native" cultures did not always understand how to live with nature better, they usually had less efficient tools to destroy it. For example a lot of the European forests were already destroyed in pre-historic times due to farming activities as well as charcoal making etc.

[matthewcerini]

To compute anything, both humans AND computers need to first translate real world quantities and relationships into their abstract mathematical representation, then apply the transformations, then translate them back to the real world. Computers have an advantage in that they don't get bogged down by keeping track of large numbers of recursive levels like humans.

That Eugenia doesn't readily acknowledge the + operator in 2+2=4 in the context of the real numbers is the obvious context in 98% of all applications and then says this is related to white Europeans using technology to intentionally destroy the planet makes me uncomfortable. That said, I agree that we have lost much indigenous knowledge, especially in growing healthy food, due to ignorance and greed (originally in the name of efficiency), not intent to harm the planet. The sad truth is that as we speak, globally, people (including people of color) are still transitioning to destructive technologies and ignorance is no longer a valid excuse.

BTW, Confidence comes from a happy childhood, trusting yourself to take risks and to go places others dare not go, and asking "clever" questions of your teachers. 😄

[funkyhairman]

I wish she would elaborate on the 2+2 logic. I really respect the social concept but I don't understand the math. How does that work in what situations. I am going to read her book maybe that will help. If there are other references, let me know.

[lawrencetorrance]

Her contention that 2 + 2 is not always 4 “to research mathematicians” boils down to saying that if you change the definition of two or plus, then the answer isn’t necessarily four. It’s a basically meaningless and I dare say disingenuous point. Anything to score a point for ideology I guess.

[joelvirolainen]

Most people understand zero when they are told that the item they bought costs all their money, and they have none left. Having no money has a very real meaning.

[NotNecessarily-ipvc]

[Monad in mathematics, science and technology]:

Monad (biology), a historical term for a simple unicellular organism

Monad (category theory), a construction in category theory

Monad (functional programming), functional programming constructs that capture various notions of computation

Monad (homological algebra), a 3-term complex

Monad (nonstandard analysis), the set of points infinitesimally close to a given point

[Monad in philosophy/cosmogony]:

Monad (from Greek μονάς monas, "singularity" in turn from μόνος monos, "alone") refers, in cosmogony, to the Supreme Being, divinity or the totality of all things.

The concept was reportedly conceived by the Pythagoreans and may refer variously to a single source acting alone, or to an indivisible origin, or to both.

The concept was later adopted by other philosophers, such as Gottfried Wilhelm Leibniz, who referred to the Monad as an *elementary particle.*

It had a *geometric counterpart, * which was debated and discussed contemporaneously by the same groups of people.

[In this speculative scenario, let's consider Leibniz's *Monad, * from the philosophical work "The Monadology", as an abstract representation of *the zero-dimensional space that binds quarks together* using the strong nuclear force]:

1) Indivisibility and Unity: Monads, as indivisible entities, mirror the nature of quarks, which are deemed elementary and indivisible particles in our theoretical context. Just as monads possess unity and indivisibility, quarks are unified in their interactions through the strong force.

2) Interconnectedness: Leibniz's monads are interconnected, each reflecting the entire universe from its own perspective. In a parallel manner, the interconnectedness of quarks through the strong force could be metaphorically represented by the interplay of monads, forming a web that holds particles together.

3) Inherent Properties: Just as monads possess inherent perceptions and appetitions, quarks could be thought of as having intrinsic properties like color charge, reflecting the inherent qualities of monads and influencing their interactions.

4) Harmony: The concept of monads contributing to universal harmony resonates with the idea that the strong nuclear force maintains harmony within atomic nuclei by counteracting the electromagnetic repulsion between protons, allowing for the stability of matter.

5) Pre-established Harmony: Monads' pre-established harmony aligns with the idea that the strong force was pre-designed to ensure stable interactions among quarks, orchestrating their behavior in a way that parallels the harmony envisaged by Leibniz.

6) Non-Mechanical Interaction: Monads interact non-mechanically, mirroring the non-mechanical interactions of quarks through gluon exchange. This connection might be seen as a metaphorical reflection of the intricacies of quark-gluon dynamics.

7) Holism: The holistic perspective of monads could symbolize how quarks, like the monads' interconnections, contribute holistically to the structure and behavior of particles through the strong force interactions.

[Shuffler..]

31:46 : "... I've used some Category Theory to understand the interactions between people. What I do think it helps with is understanding the interactions of groups of people within structures because unfortunately our society is made up of power hierarchies ..."

It looks to me as if she is trying to use Category Theory to come up with a model for _Intersectionality._
The Wokeness is strong in her.

[JamesVytas]

9:57 is the word ‘abstraction’ also ambiguous. Or does it simply mean low resolution?

[RWin-fpjn]

The subject is great. Eugenia is fun and engaging. To many, the miracle of complex numbers is what our reality is all about. Why is it that we need complex numbers to cross into the quantum world to make calculations there? It seems we don't know. But surely we must WANT to know, if we are actually interested in the the ToE that combines the physical world of relativity (the grid of space and time) to the quantum world (the grid of energy and mass). If we find the physical explanantion behind complex number then surely we have come to an end in our quest of the ToE after 100 years of failures, right? Right?
So then, let me give you the underlying physical equivalent of complex numbers; Study the nature of 'quantum leaps' of electrons in atoms and we all agree that in the world of QP the grid is defined as energy (inverse space) and the clock is defined by mass (mass equals inverse time, Sir Roger Penrose 2020, substituting E=hf into E=MC2). Consequently, in our spooky QP world, speed is then defined in [J/kg] which equals [Nm/kg] or gamma*[m2/s2] in our spacetime terms. So then, if our QP world compensates for our spacetime world, this means that for any speed in our world expressed as [m/s] we MUST have a compensating speed defined as [m2/s2] in the quantum world. Full stop! Thats impossible because [m/s] surely can never compensate for its squared value of [m2/s2] . True. Unless we insert a magic trick and use the prefix of i2=-1 in the quantum world. Then both speeds cancel always in this dual physical setup we have known now for 100 years. Thats all there is to it. So now we see. Our dual physics forces the existence of complex numbers in math. Without it, there would be no equilibrium on any scale in our cosmos. There is no other mathematical construct more important. Our good friend Isaac Newton kind of got a good start with his conservation of energy, but the true conservation Law is the one above. The grid changes the moving object and the moving object changes the grid. They don't cancel out at all. There is an equilibrium via the INVERSE dual relation of measures (mass time energy space) and our continuum functions (inertia, clock, potential, grid) with our quantum world. Math of complex numbers simply gave us the derived corresponding tool to use this truth. We were allowed to use the math, just not to allowed to see the underlying physics. I'd say after 100 years, the pivot number of 1 deserves our appreciation as well....Quite ingenious to design our world around the number one, wouldn't you say?

[joeerau]

Frankly, she was just a tad too woke for me. She went off on an unrelated tangent regarding 2+2 does not equal 4. I will never understand how mathematics, as it has involved, is any way racist regardless of who made or where a contribution has originated. If I missed her point, fine. But I still think she engaged in gobbledygook and new speak.

[Shadow_Bnned]

"Is math real?" Yes, . It's a real tool that can be used to describe real and unreal phenomena with real and imaginary numbers. Zero is an interesting number because it’s both real and imaginary. For example, you can describe the state of an object as having 0 charge. This means the object exists (it’s “real”), but no electrons are moving (hence, the current is “0”). Without the concept of zero, we wouldn’t be able to accurately describe such a state. So, in this sense, zero plays a crucial role in our mathematical descriptions of the world.

[drumhikeandski]

Her diatribe concerning no "right" answers is an abuse of what she previously condemned: she's setting herself above the audience with some implied math knowledge and saying "actI know and you dont." As a mathematician her argument is trivial. As an early comment pointed out, sure, if I change the definition of two or addition (eg mod arit, ) then the answer isn't four. Still there is a correct answer. Well, I paused the interview, so perhaps have spoken prematurely. Please Brian, be an adult an push back!

[jedser]

She lost me at ‘there’s no absolute truths in mathematics.’ Saying that 2 and 2 may not be 4 in some other math world is fine; but in the shared world in which precise definitions or values represent objects of measurement, then there are absolutely absolute truths. The world that comes to mind where 2 and 2 is not 4 is in Orwell’s 1984.

[Grey_Just_Grey]

3:50 Yeah? Well I find "Math" insulting. How do you like them apples? My understanding is that "Maths" is a contraction of "Mathemetics"; that is to say, the many diciplines of maths, hence the plural.

Granted, it is the anglicised version of the Old French "mathematique", from Latin "mathematica", from from Greek mathēmatikē, but, since we're speaking English, I'll stick with "Maths", and you can toughen up and get over your hurt feelings. 😂

[otomarjupiter]

Dear Prof. Keating, machine can represent inf similarly as human brain can. And it already does, in language of symbols and formal rules. If it complies with solid math theory it is basically indifferent from our gymnastics with inf. Even Gpt3.5 can do it pretty correctly in simpler tasks.

It is just a symbol game, domain of all languages

[גבריאל-חי]

Mathematics is the language of quantification. It is an abstraction of physical quantities and can not prove its own consistency or describe relationships that can not be quantified.

[timbranniganmusic]

Great interview, Brian! I could listen to her ideas all day. Great stuff, thanks!

[mykrahmaan]

Normally it is mentioned that zero is essential for any system of numbers with positional value.

Yet a system of numbers with positional value, but without zero as well as negative numbers, is not only possible, but essential for integrating ARITHMETIC, GEOMETRY and PARTICLE PHYSICS.

Below is the example of a sequence upto 20 of a system with positional value with base 9 without zero:

Base 10 ~ Base 9

1 1
. .
. .
. .
9 9
10 11
11 12
. .
. .
. .
17 18
18 19
19 21
2o 22

and so on.

This can be done for any base.

Even the binary system can be represented with 1 and 2 instead of 0 and 1:

1) 1
2) 2
3) 11
4) 12
5) 21
6) 22
7) 111

and so on.

The Copernican mediocrity assumption and the first law of thermodynamics prohibit the only practical possibility that TIME and LIFE be both derived (not just explained) particle physically if particles are generated at the Center of The Earth.

And these particles (generated towards other Centers around it in cubes i upto i = 3000) can NOT be represented by any arithmetic with less than 3000 DIGITS as their sizes differ non-uniformly, hence can only be represented by unique digits corresponding to particles with successive elements of non-uniform size differences among them.

Standard model as well as the periodic table are "proved" by their applicability to nonliving matter using calculations with particles assumed to have sizes in uniform multiples . But the particles that cause the DIGIT perceptions in our minds have non-uniform size differences and are multi trillions of times tinier than photons.

Hence, they cannot be represented by any mathematical formula with uniformly varying digits and no nonliving instrument can ever be sensitive to them, rendering experimental and observational science ~ that depends on "formulae with uniformly growing DIGITS", and "measurements ONLY by NONLIVING instruments" ~ a meaningless exercise for detecting life function.

In addition, the Cartesian geometry with arbitararily chosen negative and positive directions has absolutely no relevance whatsoever for particles generated at the Center of The Earth.

[sludgiebear]

I quite liked Eugenia, even though I disagree with some of the things she stated (some of which, admittedly, out of her wheel house).

ZERO INDEX
I feel the 2000/2001 millennium "topic" is an issue of argument being muddied from multiple fronts: perspective and framing of the count (very much like the "first floor"/"ground floor" issue), how we discuss the current count of the year socially, as well as education. For example: socially, we'd say: "we're *in* the year 2023", but we're actually not. Instead, 2023 full years have passed since zero and we're technically "in" 2024. It also doesn't help that our count for days and months is inconsistent to the count of the year: days and months start at 1, but our years start from zero. I feel that if we spent more time teaching our kids to better understand both the framing and inconsistencies within some of our systems - in general - we'd have less confused adults running around muddying the waters further.

I prefer zero to be the beginning point of any count because - especially in such examples - we're talking about distances, and there'll always be a distance of zero from which to begin. Differences in perspective and framing, as well as understanding, is what conjures such debates. What irks me most is that elevators/lifts don't always elevate/lift. 😆

AI, ML, LLM, ETC
If her only example of why she feels AI is not a risk is that they're bad at selecting Ads for her, then I'd urge her to transpose such an experience to the fact we continue to propel ourselves to relying upon technology and AI/ML more and more, across many different fields and implementations/focuses. We've come to heavily adopt, and we inevitably come to rely upon, technology more and more. So, imagine, for example, a surgeon relying upon AI/ML technologies during a medical procedure. Or, perhaps, like some have certainly tried to do (in the USA, at least): utilise AI/ML within legal proceedings - which have been criticised for bias. As humans, we have flaws. Do we really want to risk compounding those flaws into an AI/ML system which could dictate our livelihood, innocence, or even - dare I say it - right to exist? It's a very real risk if left without appropriate oversight, and how many of our world leaders and politicians understand even the fundamentals of such systems enough to truly comment?

POLITICS ETC IN EDUCATION
It's crazy to me that a discussion on methematics included questions based in politics and racism, but I'm not devaluing the notion because I recognise the political and tribal divides running somewhat rampant within certain societies. It's pretty sad to me, as I feel we're in somewhat of a state of social devolution, but I hold hope in that we'll surface from the next "wave" more enlightened and more wise.

REVITALISING EDUCATION
Whilst at college, I remember seeking out a physics professor to ask a question (physics was not what I was studying).The first professor I found, I politely introduced myself and asked my question, and he responded with something to the effect of: "Good question. I don't know. Let me get back to you". A short time later we saw each other and he explained what he'd researched which answered my question. I've always respected that: not only did he validate that I had a good question, he admitted he didn't know off-hand, and he took the time to research an answer for me - and I wasn't even one of his students, nor a physics student! I wish I remembered his name. I'm sorry, professor!