Tuesday, June 22, 2021

935: Maths works....

Why does math works? At the core of this mystery lies an argument that mathemati­cians, physicists, philosophers and cognitive scientists have had for centuries: Is math an invented set of tools, as Einstein be­lieved?


 Or does it actually exist in some abstract realm, with hu­ mans merely discovering its truths? 


Many great mathemati­cians, including David Hilbert, Georg Cantor and the group known as Nicolas Bourbaki, have shared Einstein’s view, associated with a school of thought called Formalism. 


But other illustri­ous thinkers, among them Godfrey Harold Hardy, Roger Pen­rose and Kurt Gödel, have held the opposite view, Platonism. 


This debate about the nature of mathematics rages on today and seems to elude an answer.


I believe that by asking simply whether mathematics is invented or discovered, we ignore the possibility of a more intricate answer: both invention and dis­covery play a crucial role.


Although eliminating the dichotomy between invention and discovery does not fully explain "the un­reasonable effectiveness of mathematics" ,


 the title of a famous article by Eugene Wigner (1960), the problem is so pro­found that even a partial step toward solving it is progress. 


In the previous lecture, I referred to the passive side of mathematics when math calculations predict empirical observations


and the active side when pure mathematics accidentally happens to be very useful in describing some feature of nature.


A pattern emerges: humans invent mathematical concepts by way of abstracting elements from the world around them, 


shapes, lines, sets, groups, and so forth, either for some specific purpose or simply for fun. 

They then go on to discover the con­nections among those concepts. 


Because this process of inventing and discovering is man­made, unlike the kind of discovery to which the Platonists subscribe, our mathematics is ultimately based on our perceptions and the mental pictures we can conjure. 


For instance, we possess an innate talent, called subitizing, for in­stantly recognizing quantity, which undoubtedly led to the con­cept of number. 


We are very good at perceiving the edges of indi­vidual objects and at distinguishing between straight and curved lines and between different shapes, such as circles and ellipses,  


abilities that probably led to the development of arithmetic and geometry. 


So, too, the repeated human experience of cause and ef­fect at least partially contributed to the creation of logic and, with it, the notion that certain statements imply the validity of others. 


This is an interesting all be it a partial answer. Mathematics isn't THE universal language, describing reality, however, in certain areas, it is a highly effective one yet. 


There exists a whole host of phenomena for which no accurate mathematical predictions are possible, sometimes not even in principle. 


In economics, for example, many variables, the detailed psychology of the masses, to name one, do not easily lend themselves to quantitative analysis. 


The predictive value of any theory relies on the constancy of the underlying relations among variables. 


Other areas are, for instance, psychology sociology, literature or art. 


Mathematicians have developed statistics and probability to deal with such shortcomings, but mathematics itself is limited, as Austrian logician Gödel famously proved. 


Mathematical systems aren't universally applicable. Take for instance integers. Perfect for counting the number of raindrops, 1,2, 3, etc. 


But what to do with that system when the raindrop falls in the ocean and you still want to continue counting?


One thing is for sure, mathematics is probably the greatest example of our ability to recognize patterns and formulate the principles behind them.


Thank you for your attention again....

MacMillan The Encyclopedia of Philosophy, 2nd edition
Routledge Encyclopedia of Philosophy, 1995
Rens Bod: "Een Wereld vol Patronen".  2019

The Discussion

[13:22] .: Beertje :. (beertje.beaumont): Thank you Herman

[13:22] Particle Physicist Bejiita (bejiita.imako):

[13:23] herman Bergson: So, in general mathematics is the result of the interaction between the mind and empirical experiences caused by a real world

[13:23] Particle Physicist Bejiita (bejiita.imako): I guess thats a good explanation

[13:23] oola Neruda: Physicists have named some of their "found concepts" Truth and Beauty

[13:24] herman Bergson: It makes me think of the epistemology of Kant (ca.1730....)

[13:24] oola Neruda: and they have a reverentail attitude toward these mysteries of ath

[13:24] oola Neruda: math

[13:25] herman Bergson: Well...ti is our mind that appears the actual creator of maths

[13:26] oola Neruda: or call them "observed concepts"

[13:26] Particle Physicist Bejiita (bejiita.imako): like the relational concept of pi

[13:26] Particle Physicist Bejiita (bejiita.imako): circum/diameter

[13:26] herman Bergson: Like Kant said.....we never can know "das Ding an Sich", the object as such....we only have our sensory experiences and the categoris of the mind, causality, space , time, that enables us to know the external world

[13:27] herman Bergson: We only can know within the categories of space and time for instance....

[13:28] herman Bergson: they create the lasting object

[13:28] herman Bergson: Maybe you disagree and take a Platonist viewpoint?

[13:29] herman Bergson: hmmm...seems you don't ^_^

[13:30] herman Bergson: When we think of the universe we only can think in terms of space and time

[13:30] Particle Physicist Bejiita (bejiita.imako): well math is very special indeed

[13:30] Particle Physicist Bejiita (bejiita.imako): and indeed also it can not be applied to eevrything, chaos is one example cause there is no patterns

[13:30] herman Bergson: then we think of atoms, neutron, protons we only can thing of mass....which must be some substance with extension in space

[13:31] Particle Physicist Bejiita (bejiita.imako): one example, when u throw the dice in the plup-game its not truly random, a computer cant randomize cause it uses math

[13:31] Particle Physicist Bejiita (bejiita.imako): it can only simulate it good enough

[13:31] herman Bergson: true....

[13:31] Particle Physicist Bejiita (bejiita.imako): using an algorithm

[13:32] Particle Physicist Bejiita (bejiita.imako): for true randomization it would require filming truly random events like snowfall or lava lamps and use data from that film stream to create true randomization

[13:33] Particle Physicist Bejiita (bejiita.imako): thats the only way for a computer to create true randomness, using external truley random data

[13:34] herman Bergson: Interesting observation....

[13:34] .: Beertje :. (beertje.beaumont): wouldn't that be an algorithm too?

[13:34] Particle Physicist Bejiita (bejiita.imako): some crypto software generate their keys with lava lamps for ect

[13:34] Particle Physicist Bejiita (bejiita.imako): a computer films lots of them and makes crypto key

[13:35] Laz Dresler is online.

[13:35] herman Bergson: Beertje, that is according the principle that what data are used as input, such dat will appear as output

[13:35] Particle Physicist Bejiita (bejiita.imako): or as we commonly say

[13:35] herman Bergson: That is what an algorithm  can handle

[13:35] Particle Physicist Bejiita (bejiita.imako): shit in =  shit out

[13:36] Particle Physicist Bejiita (bejiita.imako):

[13:36] herman Bergson: no no Bejiita....

[13:36] Particle Physicist Bejiita (bejiita.imako): eheh

[13:36] herman Bergson: the original phrase is garbage in, garbage out ^_^

[13:36] Particle Physicist Bejiita (bejiita.imako): yes i was referring to that but thats in Swedish

[13:37] herman Bergson: ^_^

[13:37] Particle Physicist Bejiita (bejiita.imako): skit in skit ut

[13:37] oola Neruda: smiles

[13:37] Particle Physicist Bejiita (bejiita.imako):

[13:37] herman Bergson: Never to old to learn (Swedish:-)

[13:37] Particle Physicist Bejiita (bejiita.imako):

[13:37] herman Bergson: But we can conlude a few things here

[13:37] CB Axel: In English it was originally GIGO.

[13:38] Particle Physicist Bejiita (bejiita.imako): yep

[13:38] herman Bergson: At least mathematics is an impressive achievement of the brain of homo sapiens in recognizing pattern and the principles behind them

[13:38] herman Bergson: Ahh, CB, nice

[13:38] oola Neruda: fractles for example

[13:38] Particle Physicist Bejiita (bejiita.imako): yes¨

[13:39] herman Bergson: A second thing is that the mathematical patterns and priciples aren't the complete tool for everything....

[13:39] herman Bergson: And what puzzles me already for a long time.....

[13:40] oola Neruda: depends upon what you are measuring

[13:40] herman Bergson: Astrophysics is an impressive theoretical science also build on mathematics....

[13:40] Particle Physicist Bejiita (bejiita.imako): yep

[13:40] oola Neruda:  but does not need to be physical

[13:40] herman Bergson: the big bang is a mathematical extrapolation of mathematical data and observations.....

[13:41] herman Bergson: but it cant answer the where/when question....

[13:41] herman Bergson: ANd I wonder why they don't conclude that this is the result of how our brain computes?

[13:42] oola Neruda: scientists want evidence

[13:42] Particle Physicist Bejiita (bejiita.imako): our view of whats around us

[13:42] herman Bergson: we can only think in space and time....and Kant said that that are our computational tools to be able to have sensory experiences that make sense

[13:43] herman Bergson: Yes, evidence,  but it is created by our own mind to some extend....so, biased evidence

[13:43] oola Neruda: yup

[13:43] Particle Physicist Bejiita (bejiita.imako): aa i guess

[13:43] oola Neruda: theory

[13:44] oola Neruda: observation

[13:44] herman Bergson: I have no problem with that.....but science sometimes has problems with the observation that some things can not be knows by us due to how our brain computes and relates to the external reality

[13:45] CB Axel: Like trying to see in 4 dimensions.

[13:45] CB Axel: I try to imagine what the 4th dimension looks like, but I just can't.

[13:45] herman Bergson: With athematics we can  however create a useful model

[13:45] Particle Physicist Bejiita (bejiita.imako): ah

[13:45] herman Bergson: Indeed CB

[13:45] oola Neruda: and look at data very carefully...

[13:45] Particle Physicist Bejiita (bejiita.imako): we would just see strange blobs

[13:45] herman Bergson: What should it look like

[13:46] herman Bergson: Here in SL we have the X, Y and Z axes and time.....

[13:46] herman Bergson: But let's not begin to philosophize about time, please ^_^

[13:46] CB Axel: I was just watching a sci-fi show where people went missing and it was discovered that they had been sucked into another dimension.

[13:46] oola Neruda: i've had physicists try to explain alternate universes to me.... it is hare to follow and they step back to call it theory

[13:46] Particle Physicist Bejiita (bejiita.imako): a good way to visualize it is to think of a 2d character on a paper with a 3d object passing through, he would only see a slice of the object changing shape as it passes through

[13:47] CB Axel: Right. Like in Flatland.

[13:47] Particle Physicist Bejiita (bejiita.imako): exactly

[13:47] Particle Physicist Bejiita (bejiita.imako): thats what flatland is all about

[13:47] Particle Physicist Bejiita (bejiita.imako): to understand why we cant see 4d

[13:48] CB Axel: I understand why we can't see it, but I'd still like to. :)

[13:48] Particle Physicist Bejiita (bejiita.imako): indeed

[13:48] herman Bergson: isn't that some novel.....about flatland.....two dimensional beings?

[13:48] CB Axel: Yes.

[13:48] herman Bergson: I have read something like that long ago

[13:49] Particle Physicist Bejiita (bejiita.imako): yes but as i get it they then encounter a 3 dimensional object, and flatland was created with this very purpose as i understand

[13:49] Marlena Rickettsi (marlenalilly): And there is a painting by Dali, something of a four dimensional cross.

[13:49] herman Bergson: I remember it was fun to read....

[13:49] Particle Physicist Bejiita (bejiita.imako): i have not read/seen it though

[13:49] Particle Physicist Bejiita (bejiita.imako): aaa Dali is cool

[13:49] Particle Physicist Bejiita (bejiita.imako): and surrealism in general

[13:50] Particle Physicist Bejiita (bejiita.imako): mysterious anc cozy in a way i like

[13:50] herman Bergson: Do you know the book CB?

[13:50] Particle Physicist Bejiita (bejiita.imako): well its dream art

[13:50] Particle Physicist Bejiita (bejiita.imako): so

[13:50] CB Axel: Flatland? Yes. I've read most of it, but I can't seem to get through the whole thing.

[13:51] herman Bergson: Well...I guess thois was a somewhat tough philosophical session today .....

[13:51] Particle Physicist Bejiita (bejiita.imako): (checking up flatland)

[13:51] .: Beertje :. (beertje.beaumont): Flatland: A Romance of Many Dimensions is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co

[13:51] herman Bergson: Ahhh....that's it!

[13:51] Particle Physicist Bejiita (bejiita.imako): ok 2d or 3dplupp today?

[13:51] Particle Physicist Bejiita (bejiita.imako):

[13:52] Particle Physicist Bejiita (bejiita.imako): we do the usual 3d version I think

[13:52] herman Bergson: Time to cool down, I think :-)

[13:52] herman Bergson: Next dimension is a pleasnat weekend, I'd say :-)

[13:52] Particle Physicist Bejiita (bejiita.imako): YAY! (yay!)

[13:52] .: Beertje :. (beertje.beaumont):

[13:53] Particle Physicist Bejiita (bejiita.imako): BEER BEER AND MORE BEER

[13:53] Particle Physicist Bejiita (bejiita.imako): yum!

[13:53] Particle Physicist Bejiita (bejiita.imako):

[13:53] bergfrau Apfelbaum: thank you! Herman and class:-)

[13:53] herman Bergson: Then, thank you all again......and see you again next Tuesday in good health, I wish ...

[13:53] CB Axel: Thank you, Herman and everyone.

[13:53] herman Bergson: Class dismissed.....

[13:53] Marlena Rickettsi (marlenalilly): thank you Professor.

[13:53] Particle Physicist Bejiita (bejiita.imako):

[13:53] .: Beertje :. (beertje.beaumont): Thank you Herman

[13:54] herman Bergson: You all survived....good ^_^

[13:54] .: Beertje :. (beertje.beaumont): i need vacation:)

[13:54] CB Axel: Me,  too. :)

[13:54] Particle Physicist Bejiita (bejiita.imako): aaa soon time

[13:54] Particle Physicist Bejiita (bejiita.imako): july for me

[13:54] Particle Physicist Bejiita (bejiita.imako): as usual

[13:55] herman Bergson: Yes....vacation is closing in on us :-)

[13:55] .: Beertje :. (beertje.beaumont): going tomorrow;)

[13:55] .: Beertje :. (beertje.beaumont): yay

[13:55] Particle Physicist Bejiita (bejiita.imako): nice

[13:55] Particle Physicist Bejiita (bejiita.imako):

[13:55] bergfrau Apfelbaum: your vacation starts tomorrow, yes?

[13:55] bergfrau Apfelbaum: ahjaa

[13:55] bergfrau Apfelbaum: yay

[13:55] .: Beertje :. (beertje.beaumont): yes

[13:55] CB Axel: Are you going somewhere nice?

[13:55] .: Beertje :. (beertje.beaumont): yes yes:)


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