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There's more to mathematics than rigour and proofs (2007)

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Re: There's more to mathematics than rigour and proofs (2007)

#91

Post by ocodo »

karmakurtisaani wrote: post #58
FrustratedMonky wrote: post #38

Earlier quoted context omitted.

man, sometimes things are obvious and right in front of us. I've known the Dogen saying for years. Have even meditated on it. I've been enjoying Bell Curve meme for sometime. Very funny. Never put together that these were the same thing. Today I am awakened.

At first, the Dogen saying and the bell curve meme seem like different things, then..

first there is the bell curve,

then there is no bell curve,

then there is the bell curve...

If you learn anything significantly deeply, this is a repeating pattern.

Re: There's more to mathematics than rigour and proofs (2007)

#93

Post by Tainnor »

JoeyBananas wrote: post #86
red_trumpet wrote: post #45

Earlier quoted context omitted.

Yeah, they should have heard about ZFC and have a notion what a formal proof is. On the other hand, I'm not sure your last sentence is really that relevant. > They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff I'm doing a PhD in algebraic geometry, and that stuff isn't relevant at all. To me "everything is a set" pretty much applies. Hell,…

> I'm doing a PhD in algebraic geometry, and that stuff isn't relevant at all. Yes, exactly. These are topics that 99% of legit mathematicians don't know or care about. It's like saying "I'm a computer expert" when you only know Python, and then a computer engineer that designs CPUs starts laughing at you

I don't understand the point of your comments.

Re: There's more to mathematics than rigour and proofs (2007)

#94

Post by empath75 »

auggierose wrote: post #78
tsaixingwei wrote: post #22

“Before I learned the art, a punch was just a punch, and a kick, just a kick. After I learned the art, a punch was no longer a punch, a kick, no longer a kick. Now that I understand the art, a punch is just a punch and a kick is just a kick.” - Bruce Lee

You can see people go through that process right here on HN, slowly realizing that 1 the integer is the same as 1 the real number.

That thread felt like I was talking crazy pills. So many people confused by the difference between the construction of numbers using some particular set of foundational axioms and the properties of numbers that should hold true _regardless of the constructions_. Obviously the "integer 1" is not strictly speaking "the same as" the "rational number 1" when constructed in set theory, but there's a natural embedding of the integers into the rationals that preserves all the essential properties of the integer 1 when it's represented as the rational number 1. Confusing the concept with the encoding, basically.

Re: There's more to mathematics than rigour and proofs (2007)

#96

Post by JoeyBananas »

Tainnor wrote: post #93
JoeyBananas wrote: post #86

Earlier quoted context omitted.

> I'm doing a PhD in algebraic geometry, and that stuff isn't relevant at all. Yes, exactly. These are topics that 99% of legit mathematicians don't know or care about. It's like saying "I'm a computer expert" when you only know Python, and then a computer engineer that designs CPUs starts laughing at you

I don't understand the point of your comments.

My claim is that mathematics as practiced by mathematicians is not as rigorous as they think it is, and infact they're not even aware of the advances in rigorous mathematics that they're not using. Even though those advances are super important.

Re: There's more to mathematics than rigour and proofs (2007)

#97

Post by topologie »

rokob wrote: post #19

> The distinction between the three types of errors can lead to the phenomenon ... of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors I was initially amazed at this when I was in graduate school, but with enough experienc…

Good point. Here are some notes on it based on what I've observed happens in Academia and in other environments:

I think handwaving comes in different flavors:

- Handwaving and not knowing what they are doing, when they know they don't know:

This is arrogance and/or fear of people thinking you are a fool. Bad practice. Professionals who do this are status chasers and not fun to be around. Students who do this are mostly insecure, and they might just need some help with their self-esteem. Help them by letting them feel comfortable with being wrong. Foster a good environment so that the arrogance and fear fade away.

- Handwaving and not knowing what they are doing, when they don't know they don't know:

I believe this is a good thing, in particular for Students, if they are within a nurturing environment. It can lead to interesting ideas and to discussions of innovative ways to move forward. I believe this to be a way of actually "training your intuition muscle" both for Students and Professionals. It lets them know not to fear moving on, tackling the thing that captures their attention the most at first, and later on filling some of the gaps, which I feel is common practice for people who have been working on the field for a while. However, if the gaps are left unattended it can lead to bad things... Environment matters.

- Handwaving and knowing what they are doing, when they know they don't know:

For trained Professionals only... :) This modality kinda kicks in when deep in mathematical work. It's the path that leads to the Eureka moments... Pure trained intuition acting almost as a separate entity to oneself. We are facing the unknown and something tells us that certain aspect can be handwaived, we don't fully know why but we feel it is. Later on it becomes clear why we could do the handwave. It works itself out.

- Handwaving and knowing what they are doing, when they don't know they don't know:

For trained Professionals only... Kind of a stretch, but might be where our intuition either fails us completely, or completely takes us by the hand to turn the unknown unknowns into known unknowns, then it goes back to the previous category.

This isn't set in stone by the way, just some thoughts I had while reading the article...

Any ideas or suggestions for modifications more than welcomed.

Re: There's more to mathematics than rigour and proofs (2007)

#98

Post by topologie »

srik wrote: post #34
tsaixingwei wrote: post #22

“Before I learned the art, a punch was just a punch, and a kick, just a kick. After I learned the art, a punch was no longer a punch, a kick, no longer a kick. Now that I understand the art, a punch is just a punch and a kick is just a kick.” - Bruce Lee

For the interested, the original Dōgen zen koan goes something like this — Before I began to practice, mountains were mountains and rivers were rivers. After I began to practice, mountains were no longer mountains and rivers were no longer rivers. Now, I have practiced for some time, and mountains are again mountains, and rivers are again rivers.

I believe this is also tied to La Subida del Monte Carmelo from San Juan de la Cruz. I'm oversimplifying, but basically it goes like this:

As San Juan climbs Monte Carmelo, he finds nothing at the base of the monte, then he finds nothing at the middle, but then, at the cusp, he finds Nothing (capital N Nothing).

See the following image for reference: https://commons.wikimedia.org/wiki/File:Monte_Carmelo_Juan_d...

There's quite a bit of parallels between proper "Catholic Mystics" and Zen teachers...

I highly recommend both the San Juan de la Cruz works, in particular Ascent of Mount Carmel and The Dark Night, along with The Cloud of Unknowing, which was an inspiration for him.

For those curious about learning more about Koans, I cannot recommend this other book highly enough: https://www.amazon.com/Two-Zen-Classics-Gateless-Records/dp/...

The book contains a collection of Koans along with Mumon's (et al) Commentary, Mumon's Verse, as well as modern day notes that help us understand some of the concepts hidden behind what looks like "poetical nonsense" at first, as well as giving the context for historical and mythological figures that are mostly unknown for those "outside the loop." What I love about the notes is that they still leave you with the opportunity to explore the koan further, properly, so they don't really take away all the fun.

Example Koan from the book:

##########################################################################

Case 4 The Western Barbarian with No Beard

Wakuan said, "Why has the Western Barbarian no beard?"

Mumon's Comment:

Study should be real study, enlightenment should be real enlightenment. You should meet this barbarian directly to be really intimate with him. But saying you are really intimate with him already divides you into two.

Mumon's Verse:

Don't discuss your dream before a fool. Barbarian with no beard Obscures clarity.

NOTES (abridged)

- The Western Barbarian: The Western Barbarian stands for Bodhidharma, who brought Zen to China from India. He is always depicted with a beard. The case therefore means, "Why doesn't Bodhidharma, who has a beard, have no beard?"

- Meet this barbarian directly: This is not really a meeting but a becoming. You should yourself become Bodhidharma. Then if you have a beard, Bodhidharma has a beard; if you have no beard, then neither had Bodhidharma. But can you say that you are in truth Bodhidharma?

[...]

- Obscures clarity: Words, concepts, and other inventions of the mind only obscure the truth. Do not cling to shadows, but catch hold of truth itself.

##########################################################################

You get the idea.

Enjoy!

Re: There's more to mathematics than rigour and proofs (2007)

#99

Post by Tainnor »

JoeyBananas wrote: post #96
Tainnor wrote: post #93

Earlier quoted context omitted.

I don't understand the point of your comments.

My claim is that mathematics as practiced by mathematicians is not as rigorous as they think it is, and infact they're not even aware of the advances in rigorous mathematics that they're not using. Even though those advances are super important.

What important discoveries have you made using your supposedly "more rigorous" mathematics?

Re: There's more to mathematics than rigour and proofs (2007)

#100

Post by JoeyBananas »

Tainnor wrote: post #99
JoeyBananas wrote: post #96

Earlier quoted context omitted.

My claim is that mathematics as practiced by mathematicians is not as rigorous as they think it is, and infact they're not even aware of the advances in rigorous mathematics that they're not using. Even though those advances are super important.

What important discoveries have you made using your supposedly "more rigorous" mathematics?

Loaded question
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