Nobel Laureate mathematician, Roger Penrose, answers,

https://youtu.be/ujvS2K06dg4

I don't have time for the video... was it Joseph Smith who did it? In the study? With a candlestick?

It’s discovered.

After 40 years living with one mathematician and having collaborated with her to produce another mathematician (my elder daughter), that's what I would have said too :-)

And although I'm NOT a mathematican myself, I would also have said that humans do invent the means and techniques to discover mathematics :-)



Edited 2 time(s). Last edit at 05/08/2022 02:08PM by Soft Machine.

Soft Machine Wrote:
-------------------------------------------------------
> After 40 years living with one mathematician and
> having collaborated with her to produce another
> mathematician (my elder daughter), that's what I
> would have said too :-)
>
> And although I'm NOT a mathematican myself, I
> would also have said that humans do invent the
> means and techniques to discover mathematics :-)

Cool. My dad was a mathematician. I'm not a mathematician either, but I find this master mathematician's ideas completely fascinating, especially as they relate to Einstein's theory of singularity.

He's a genius and a fountain of knowledge about the universe, that Stephen Hawking drank from, hence Penrose–Hawking singularity theorems that just earned him a Nobel Prize in physics.

https://en.wikipedia.org/wiki/Penrose%E2%80%93Hawking_singularity_theorems

He's also a philosopher and has written amazing books on consciousness, which I have yet to delve into. I'm still wading through Road to Reality, which is blowing my mind.

https://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311/ref=asc_df_0679776311/

He talks a lot in there about Lambda, the universal constant Einstein famously called his biggest mistake, which now forms our standard model of the universe.

"After 40 years living with one mathematician and having collaborated with her to produce another mathematician (my elder daughter), that's what I would have said too :-)"

COMMENT: "collaborated"? That sounds very 'formal' :-)
_______________________________________________

"And although I'm NOT a mathematician myself, I would also have said that humans do invent the means and techniques to discover mathematics :-)"

COMMENT: Hummmmm. So then, was 'the calculus' discovered, or is it merely a means and technique that humans invented to discover mathematics? If only the means, just what 'real' mathematics did it assist us in discovering? In other words, at what point do we finally get to the platonic reality of mathematics, as Penrose expressly called it?

Mathematics and the technique of mathematics cannot be separated! The mathematical relationships are necessarily embedded in such means and techniques.

When physicists say that they 'use mathematics' to understand nature, they seem to be talking about something external to nature. What then is the relationship between mathematics and nature? Answer: Mathematics must be a transcendent, non-physical reality. If so, what other transcendent realities might there be? Where does that leave physicalism? Where does that leave science? Inquiring minds want to know!

Calculus like Statistics are a set of tools used to talk about the natural world.

A very simple example is average or mean. This is not some sort of natural law at work here, just over time people discovered that it was a great way to show central tendencies.

Like any other science natural things happen, humans discover how it works, and invent ways to apply it to make it useful.

Northern_Lights Wrote:
-------------------------------------------------------
> Calculus like Statistics are a set of tools used
> to talk about the natural world.
>
> A very simple example is average or mean. This is
> not some sort of natural law at work here, just
> over time people discovered that it was a great
> way to show central tendencies.
>
> Like any other science natural things happen,
> humans discover how it works, and invent ways to
> apply it to make it useful.

There IS a natural law at work in your example, though. For instance, the weight of gray wolves varies, but if you measured every gray wolf on the planet you would find that they form a distribution with most of the wolves close to or at the average weight and fewer of them outlying. The concept--that an average is a characteristic of a population--is discovered, but the tools of representing it graphically or calculating it from sampling data are invented.

Was this before or after they started mixing letters in with the numbers? I thought math went all to Hades when this happened.

At least the God of mathematics won't send you to outer darkness for getting the math wrong.

bradley Wrote:
-------------------------------------------------------
> At least the God of mathematics won't send you to
> outer darkness for getting the math wrong.


God is a mathematician, according to Kaku,

https://www.youtube.com/watch?v=jremlZvNDuk



Edited 1 time(s). Last edit at 05/08/2022 05:21PM by schrodingerscat.

Didn't Al Gore invent everything?

Just the Internet. I was watching the news when he made the claim and thought it was a really dumb thing to say even if by some long stretch it were true.

bradley Wrote:
-------------------------------------------------------
> Just the Internet. I was watching the news when he
> made the claim and thought it was a really dumb
> thing to say even if by some long stretch it were
> true.


Not quite as dumb as claiming Al Gore said he invented the internet,

https://www.snopes.com/fact-check/internet-of-lies/

or as dumb as Putin's Puppet saying "nobody has done more for Christianity, or for evangelicals, or for religion itself than me."

https://www.newsweek.com/donald-trump-claims-nobody-has-done-more-religion-itself-him-1635036



Edited 3 time(s). Last edit at 05/08/2022 03:09PM by schrodingerscat.

I most of the time fall into the "math is discovered" camp.

However.....

Just to take one example that most people have had some minor exposure to, some people figured out at least 4,000 years ago that if you square a number, it doesn't matter if the number is positive or negative, the square will be positive.

The Babylonians in 1,800 BCE already had derived the quadratic formula (and how to calculate the square root of 2, and knew about what we now call Pythagorean triangles, and more - they really kicked ass mathematically) and came up with situations where the square root value in the quadratic formula was negative. They basically assumed that was nonsense, and ignored those situations. They knew there was no number, that when you squared it, would give a negative result. Can't get there from here.

Over the centuries, other situations arose in math where the solution would be a number, which when squared, was negative. It finally got so hard to ignore, that mathematicians basically decided to pretend that there was a number, which when squared, is negative. They decided early on to name this value "i", the first letter in the word "imaginary" in various European languages.

Imaginary and complex numbers turned out to be immensely useful in applied mathematics, electrical engineering in particular.

It's really hard for me to convince myself that "i" was not invented. I mean, just the name "imaginary" tells you what mathematicians thought about it. They "knew" no actual number squares to a negative value

About the best I can do is say that it was an invention of something that was actually there in mathematics waiting to be discovered, but it was invented first, and the discovery came later.

For example, electrical engineers don't think of i as imaginary at all. It is the heart and soul of digital signal processing. It represents an orthogonal ("at a right angle to") part of a digital signal, and neither part of the signal is any more real or imaginary than the other. They are just two aspects of the same signal.

EEs don't even use the letter "i" for imaginary numbers. It was already used to represent something to do with electricity, so they use j to represent imaginary numbers. Confuses math majors all to hell at first.

Is math a language?

I think it is. I think it's a about patterns and that is why it matches up well with science. What we call science is about repeated observations or patterns in nature. So to use a pattern language to describe and manipulate patterns is very powerful.

I would also use Goedel as a proof of its being a language. That math is incomplete. That there are propositions that are true that cannot be proven. The whole self referential problem at the base of set theory. They ended up just having to redefine it away to "solve" the problem.

Not only is it a language, it is a language that has been adopted universally around the world. The pronunciation is different in different base languages, but the written form is the same everywhere as far as I know.

It is a way of describing relationships and patterns, and can be far more efficient and effective at doing that than text.

Thank goodness the Arabs invented a better number system. Working with Roman numerals sucks but they do look cool.

I think it was from India.

Should arabic numbers be taught in schools?

https://www.independent.co.uk/news/arabic-numerals-survey-prejudice-bias-survey-research-civic-science-a8918256.html

Just a bunch of idiots who don’t understand they have been using Arabic numbers their whole lives. Well rounded intelligent people are a small fraction of society. The majority are quite stupid and this is why 1% own most the world’s wealth. It’s easy to play people.

Knowing the origin of "arabic" numerals is irrelevant to either intelligence, or being well-rounded. And the origin is India, not the Middle East.

On top of which, the Babylonians 3,800 years ago had a nearly identical system to the arabic system of written numeration except the shapes for the individual digits were quite different. It took them several centuries longer to come up with a symbol for zero. The key development was that they did not invent new symbols for larger and larger numbers, like the Greeks and Romans did, they used the same symbols that were used for units, and the position in the number told them whether a 2 was 2 or 20 or 200. They also handled fractions the way we do. Greek, Roman and Egyptian fraction rules were a mess. Even people who still think Roman numerals are cool, can't do Roman fractions.

"It's really hard for me to convince myself that "i" was not invented. I mean, just the name "imaginary" tells you what mathematicians thought about it. They "knew" no actual number squares to a negative value."

COMMENT: The statements of Penrose are interesting here. As a reminder, Penrose is expressly a member of the discovery (as opposed to the 'invented') camp--including, perhaps most especially imaginary and complex numbers. Here is what he said:
________________________________________________

"Likewise, the very system of complex numbers has a profound and timeless reality which goes quite beyond the mental constructions of any particular mathematician. . . ."

"While at first it may seem that the introduction of such square roots of negative numbers is just a device -- a mathematical invention designed to achieve a specific purpose -- it later becomes clear that these objects are achieving far more than that for which they were originally designed. As I mentioned above, although the original purpose of introducing complex numbers was to enable square roots to be taken with impunity, by introducing such numbers we find that we get, as a bonus, the potentiality for taking any other kind of root or for solving any algebraic equation whatever. Later we find many other magical properties that these complex numbers possess, properties that we had no inkling about at first. These properties are just *there.*"

"Is mathematics invention or discovery? When mathematicians come upon their results are they just producing elaborate mental constructions which have no actual reality, but whose power and elegance is sufficient simply to fool even their inventors into believing that these mere mental constructions are 'real'? Or are mathematicians really uncovering truths which are, in fact, already 'there' -- truths whose existence is quite independent of the mathematicians' activities? I think that, by now, it must be quite clear to the reader that I am an adherent of the second, rather than the first, view at least with regard to such structures as complex numbers and the Mandelbrot set."

(Roger Penrose, The Emperor
s New Mind, p. 125-126)

"There is something absolute and 'God-given' about mathematical truth. This is what mathematical Platonism . . . is about. Any particular formal system has a provisional and 'man-made' quality about it. Such systems indeed have very valuable roles to play in mathematical discussions, but they can supply only a partial (or approximate) guide to truth. Real mathematical truth goes beyond mere man-made constructions." (Ibid. p. 146)

Penrose's Platonic view of mathematics ties into his view of consciousness, and his rejection of mathematical insight (and cognition generally) as simply the computations of the brain. He thinks (and I agree) that there is something much deeper going on.
_______________________________________________

"About the best I can do is say that it was an invention of something that was actually there in mathematics waiting to be discovered, but it was invented first, and the discovery came later."

COMMENT: The suggestion that something was "there" waiting to be discovered, but had to be invented first, begs the question as to who invented it before human discovery? Penrose viewed mathematical reality not as 'invented' but as a Platonic reality existing as part of the universe itself. Thus, he would never say (I don't think) that some intelligent agency must have invented it prior to human discovery.
_____________________________________________

For example, electrical engineers don't think of i as imaginary at all. It is the heart and soul of digital signal processing. It represents an orthogonal ("at a right angle to") part of a digital signal, and neither part of the signal is any more real or imaginary than the other. They are just two aspects of the same signal.

COMMENT: But this is typical of engineering where the practical just *is* the reality, without a deep dive into metaphysics--including the metaphysics of imaginary and complex numbers. At bottom, they are dealing with quantum *reality*, which transcends any practical applications of quantum mathematics.

I agree with Penrose's take on imaginary numbers.

>"While at first it may seem that the introduction of such square roots of negative numbers is just a device -- a mathematical invention designed to achieve a specific purpose -- it later becomes clear that these objects are achieving far more than that for which they were originally designed.

I said I view it as originally an invention (device is there word he used) to get around a rock in the road of mathematics - the inability to take square roots of negative numbers.

We later discovered all the other nifty properties that were inherent in this device, and it seems those properties were always there, so it makes sense to say they were discovered. But "i" still feels like initially an invention, or even a dodge, to me.


Same thing happened with people trying to prove Euclid's parallel postulate by changing it in the hope of coming up with inconsistent geometries, thereby proving that the original parallel postulate was a necessary part of geometry. Only problem was, they failed to find inconsistencies, and non-Euclidian geometries were born. That was not the original goal. Throwing out the parallel postulate was a dodge to solve a problem - how do we prove the parallel postulate is necessary. The dodge failed and the problem was shown not to be a problem at all. The parallel postulate is not necessary. That was a discovery.

>Thus, he would never say (I don't think) that some intelligent agency must have invented it prior to human discovery.

Not sure what you mean by that. I consider "i" to be a human invention, not something created by "some intelligent agency", just us.

Same thing happened with people trying to prove Euclid's parallel postulate by changing it in the hope of coming up with inconsistent geometries, thereby proving that the original parallel postulate was a necessary part of geometry. Only problem was, they failed to find inconsistencies, and non-Euclidian geometries were born. That was not the original goal. Throwing out the parallel postulate was a dodge to solve a problem - how do we prove the parallel postulate is necessary. The dodge failed and the problem was shown not to be a problem at all. The parallel postulate is not necessary. That was a discovery.

COMMENT: I like this analogy and would add that not only could they not find inconsistencies when eliminating the parallel postulate, they found internally consistent alternative mathematical geometries beyond the highly intuitive Euclidian geometry. And even more startling, they later found that non-Euclidian geometry was part of physical reality, most notably within Einstein's theory of General Relativity. So, it would appear that non-Euclidian geometry was indeed discovered, not invented; i.e. that is, it was there all the time in both mathematics and physics.
____________________________________________

>Thus, he would never say (I don't think) that some intelligent agency must have invented it prior to human discovery.

Not sure what you mean by that. I consider "i" to be a human invention, not something created by "some intelligent agency", just us.

COMMENT: Penrose would disagree. He thinks 'i' (and complex number theory) was discovered, both in mathematics and physics, much like non-Euclidian geometry. That means 'i' (and complex numbers) had to be 'there' prior to discovery. But where? This is where the metaphysics of Platonic reality comes in, presumably some sort of transcendent reality that is discovered and not invented. That is Penrose's view, and it follows from the general view that mathematics is discovered and not invented. If it was discovered, then it pre-existed that discovery. And it pre-existed in some sort of Platonic realm, which is an inherent feature or property of the universe.

This 'transcendent' quality of mathematics is reminiscent of the 'transcendent' quality of consciousness and mind. In Penrose's view (as widely stated) both mathematics and consciousness represent transcendent realities in some sense independent of the physical world. Here is another quote:

"Mathematics itself indeed seems to have a robustness that goes far beyond what any individual mathematician is capable of perceiving. Those who work in this subject, whether they are actively engaged in mathematical research or just using results that have been obtained by others, usually feel that they are merely explorers in a world that lies far beyond themselves -- a world which possesses an objectivity that transcends mere opinion, be that opinion their own or the surmise of others, no matter how expert those others might be."

(See Penrose, Road to Reality, Chapter 1, particularly p. 13)

"Not sure what you mean by that. I consider "i" to be a human invention, not something created by "some intelligent agency", just us."

That is a matter of opinion, but I'm not sure Leonhard Euler would agree as he coined the term "Transcendental functions". Euler seems to have been a Platonist along the lines of Penrose and Einstein.

They may had a grasp of divinity that we little people can never achieve, so for us there is religion. For real simpletons, there is Mormonism. Can we do without a language for living that delves into platonic realities? There are a lot of pitfalls to avoid. What I see is that the language is secularized. Marvel's MCU is the new gospel stories. Life is the hero's journey looking for a tale.

It's interesting to note that this platonic realm of mathematics wasn't plumbed in ernest until its competition was killed off by the reformation, or until God was dead as Nietzsche wrote. It could be that we can never get away from whatever divinity is because we are that in human form. The Greek and Norse gods are as real as the Judaic and Mormon gods are as real as the platonic forms of mathematics, language, and anything that can be imagined.

To wrap up with 'i', we may be the imaginations of ourselves, but we are real. We are the Rasta I in i. Created in the image of Jah. Oh, I hope you like jammin, hope you like jammin too.



Edited 2 time(s). Last edit at 05/14/2022 03:27AM by bradley.

It's been years since I took a math class in college but as I recall modern math (calculus) or rather the foundations for calculus was invented by sir Isacc Newton, when he was a young man studying in boarding school he stayed over the Christmas holiday instead of going home, during this time he came up with the rules of exponents the idea that if we times two variables together the exponents would be summed or the difference taken, this concept was invented, not discovered. Its the language of math which mathematicians agree to follow, which led to Simpsons rule and Eulers formula which states something to the effect that the summation of X to the i as i approaches infinity can also be calculated or approximated using integration (the adding and subtracting of exponants) which is the basics of calculus, also known as the area under the curve. Correct me if I forgot anything.

The part of math that's made out of nowhere is the imaginary numbers part. When Issac Newton was using the ground rules for exponents everyone was in disagreement of how to calculate imaginary ideas such as the square root of negative one, so imaginary numbers were imagined up, which opened the gateway of functional calculus for a variety of unsolvable problems, and calculus helped the foundation for modern science. So the made up portion of math is imaginary numbers and the multiplying and adding of them not so much exponents themselves.

I misplaced another post here, replying to Henry, but I might as well use the space to reply to Maca. I often disagree with Maca, usually at the "bangs head on table" level of disagreement. There is one serious error in the post above, but for someone who doesn't work with math on an ongoing basis, much of it was at least in the ballpark. It is better than I would expect from the average person on the street.

First, the error. The laws of exponents were well known before the time of Newton. Multiplication is really just a faster way to write and perform repeated additions. Integer exponentiation is just a faster way to write and perform repeated multiplications. Fractional exponents involved taking roots of numbers, but that was also well known long before Newton. The laws of exponents were not new with him.

That said, differential, and later integral calculus do involve doing many tricks, both plain and fancy, with exponents, and infinite polynomials, like Taylor and MacLaurin series, which made much of the engineering of the industrial revolution possible. So, in the ballpark. Calculus was a transformational discovery.

BTW, in the English-speaking world, Newton gets the credit for calculus. In the German speaking world, Leibniz gets the credit. During their lifetimes, both accused the other of plagiarizing calculus, and they went to their graves thinking they had been wronged.

Historians since then have come to the conclusion that they both more or less simultaneously and independently invented calculus. The subject was "in the air" at the time, ripe to be developed. The notation that we use for calculus today was that that was developed by Leibniz.

Last item - I believe Newton took an entire year off from university because of Black Plague, and that was when he developed calculus in his ample spare time. I got good at sudoku during the pandemic. :-/



Edited 2 time(s). Last edit at 05/12/2022 03:07PM by Brother Of Jerry.

I wonder what this man would say?

https://news.yahoo.com/chinese-math-genius-takes-1-202005582.html



Edited 1 time(s). Last edit at 05/12/2022 03:23PM by Elder Berry.

Edited 1 time(s). Last edit at 05/12/2022 03:46PM by anybody.

Absolutely!!!

discovered or invented ?


You can make that same stupid "argument" for just about anything.

Dave the Atheist Wrote:
-------------------------------------------------------
> discovered or invented ?
>
>
> You can make that same stupid "argument" for just
> about anything.


Calling Penrose’s argument ‘stupid’ says everything I need to know about you.

Mathematics is a language so it was invented. The observations in the universe that are converted to units of measurement to be used in the mathematical language are discovered.

"Mathematics is a language so it was invented. The observations in the universe that are converted to units of measurement to be used in the mathematical language are discovered."

COMMENT: Languages are symbolic, representational systems 'invented' by human beings to facilitate communication (and thought). In this sense, mathematics is certainly a language invented by human beings, as you insist. But this is a bit trivial, and certainly not the end of the story.

In natural (written) languages, like English, the symbols are the letters of the alphabet, which are combined by convention into words, sentences, etc. In addition, in order to be effective, the symbols of the language must have 'meaning,' that is they must 'represent' something. The symbols combine to 'represent' some thing in the real world, or otherwise some abstraction that is connected to the real world in some way. Thus, 'tree' means tree, and 'unicorn' means 'that fanciful horse-like animal with a horn.'

In mathematics, the symbols are the conventional number symbols, and the various relational symbols (e.g. +,-, and x) that are associated with the language of mathematics. But, as in a natural language, these symbols must have meaning in order to be effective as components of communication. In other words, they have to represent something. What is it that pure mathematics as a language represents, in the same way that a natural language represents the 'objects' in the real world, or the 'objects' that make up our common perceptions of a real world?

This is where a platonic reality comes in. Mathematics proper (the abstract numbers themselves and their remarkably complex unified and orderly relations) transcends the language of mathematics which merely facilitates discussion of whatever it is mathematics is about or represents. Thus, mathematics proper is NOT invented, but exists independent of the language of mathematics, which was invented in order to merely have access to, and communicate about, the platonic world of mathematics proper, whatever that is.

Note, that you cannot say that the language of mathematics is just about the quantitative relations of the physical world for two reasons: First, pure mathematics precedes mathematical physics and exists separate from mathematical physics; and second pure mathematics involves logical relations and systems that are not necessarily represented in the physical world.

I agree with Penrose, math is inherent in nature.
That seems self evident everywhere you look in nature, from the singularity at the center of every galaxy, recently proven to exist by Penrose, 100yrs after Einstein calculated their existence, which earned Penrose a Nobel Prize, to the mathematical arraignment of atomic elements, molecules and fractal forms found in nature. Ferns were using fractal geometry long before Mandelbrot developed fractal geometry. Viruses were making the most complex platonic forms, long before Buckminster Fuller was inspired by them to create geodesic domes.

"I agree with Penrose, math is inherent in nature. That seems self evident everywhere you look in nature, from the singularity at the center of every galaxy, recently proven to exist by Penrose, 100yrs after Einstein calculated their existence, which earned Penrose a Nobel Prize, to the mathematical arraignment of atomic elements, molecules and fractal forms found in nature."

COMMENT: First, nobody has 'looked at' the "singularity at the center of every galaxy, or any galaxy, and found mathematics. Singularities are not empirical facts (they are not subject to experience) they are theoretical, mathematical constructs.

Second, Penrose did not *scientifically* prove that such singularities exist. What he *did* show (with important contributions from Stephen Hawking) is that singularities are a mathematical consequence of Einstein's theory of general relativity. In other words, this was a 'mathematical' 'proof' of singularities, not a scientific proof.

(As an aside, it seemed unusual for Penrose to receive the Nobel prize for a theoretical contribution without substantive physical evidence for infinite singularities. Einstein, for example, never received a Nobel Prize for Relativity presumably because it was deemed only theoretical! His 1921 Nobel was for the photoelectric effect as related to his 1905 paper on that topic, which was experimentally confirmed.)

Moreover, singularities in this sense are mathematically infinite. There are no known infinities in nature, i.e. the physical world, so there is a disconnect between general relativity and genuine 'physical' singularities. It is well known that General relativity *breaks down* at the level of quantum mechanics--the realm of singularities. As such, the relationship between singularities (if they actually exist in nature) and general relativity remains obscure. The effort to resolve this obscurity is called Quantum Gravity. (See, Lee Smolin, *Three Roads to Quantum Gravity* (2001). This is all well known, and undisputed. Penrose commented:

"At least, what we *do* know is that, so long as Einstein's picture of a classical spacetime can be maintained, acting in accordance with Einstein's equation . . . then a spacetime singularity will be encountered within the hole. The expectation is that Einstein's equation will tell us that this singularity cannot be avoided by any of the matter in the hole and that the 'tidal forces' . . . will diverge to infinity. . . . In fact, it seems unavoidable that the realm of *quantum gravity* . . . will be entered , so that these expectations of the classical theory will have to be modified in accordance with this. We do not yet know what the correct 'quantum-gravitational' theory must be, but these black-hole considerations supply us with an important input; and this input should be guiding us in the appropriate directions in our search for the correct 'quantum gravity.' " (Penrose, Road to Reality, p. 713)

Quantum theory involves discrete (quantum) entities, not infinities! Thus, the 'modification' Penrose contemplates for quantum gravity is to eliminate the infinite singularity within the center of black holes. Lee Smolin's comment is very interesting in this regard:

"Quantum gravity should have something to say about the singular region in the interior of a black hole, in which the density of matter and the strength of the gravitational field becomes infinite. There are speculations that quantum effects will remove the singularity, and that one consequence of this may be the birth of a new universe inside the horizon." (Tree Roads to Quantum Gravity, p. 192-193)

_______________________________________________

"Ferns were using fractal geometry long before Mandelbrot developed fractal geometry. Viruses were making the most complex platonic forms, long before Buckminster Fuller was inspired by them to create geodesic domes."

COMMENT: Neither ferns or other non-intelligent organisms in nature "use" fractal geometry or mathematics generally simply because their structure, form, and related physical processes can be described mathematically. Mathematics is a tool for the physicist, but it is not a tool of nature itself. (At least it seems odd to think of it that way.) The relationship between mathematics proper (i.e. the platonic metaphysical essence of mathematics) and nature is unknown, other than the exquisite correlation of nature with mathematics.

In my view, the fact that mathematics is (arguably) a transcendent, metaphysical reality precludes understanding its connection with the physical world. This is reminiscent of questions as to the relationship of mind and brain. As long as mind (consciousness) remains a transcendent, metaphysical reality, the nature of its relation to the brain will be unknown. (I think Penrose would agree with this statement, but am not sure.)

If quantum mechanics is actually correct, would suggest discovered

Math, at least the knowledge of, is continually being developed - widened, which is similar to being Discovered and invented

( aren’t all inventions discoveries)?

Schroedie, Math is the Bandersnatch, with razor-sharp little teeth that grab and claws that catch . . . Or something like that.

Nobody invented nor discovered it. It belched up from some tar pit and came after us.

Which is why I don’t work on Wall Street.

Math, to some degree is an artificial construct; if humans didn’t recognize it, it wouldn’t be in our minds/ thoughts;

Therefore math is an artificial, contrived invention, it doesn’t exist in nature at least the way we view - understand it.