George Lakoff - Is Mathematics Invented or Discovered?

Mathematics describes the real world of atoms and acorns, stars and stairs, with remarkable precision. So is mathematics invented by humans just like chisels and hammers and pieces of music? Or is mathematics discovered—always out there, somewhere, like mysterious islands waiting to be found? Whatever mathematics is will help define reality itself.

[siriboonkotchaseth]

I don't know anything about math because I am from the opposite field of sciences; social science. But one thing I see from the several lectures of Prof. Lakoff, he is a genuine, and not a snob or so arrogant scholar. He is the real kind teacher. I love how he gives lectures. This is a respect from Thailand.

[williamwolfe]

Very nice explanation of negative x negative -- as rotations -- most insightful I've ever heard.

[wade]

One of the better explanations in this series.

[joaodenardi]

Awesome explanations, I really wish that more mathematicians knew how to articulate their discipline with linguistic and social sciences in the epistemological level like Dr Lakoff does...

[robertabrown]

In most pre-college classrooms, the word metaphor is a word that describes something in terms of something else. I can understand that mathematics puts quantities in terms that add a new understanding to their relation with other quantities, which means that it uses numerical and other symbolic metaphors. So mathematics in that sense puts quantities and their relationships with one another into a form that makes them easier to manipulate in our minds, so it is a discovery that helps humans to make new discoveries about our existence. For instance, we did not invent gravity, but calculating the gravitational constant makes it easier for us to understand how and why planets and galaxies, and falling apples, do their thing!

[Mephistel]

Thank you for introducing me to George Lakoff and his body of work.

[yourkingdomcomeyourwillbedone]

That part at 8:17 was brilliant. I’ve had this thought before but couldn’t articulate it quite like George did.

[drbonesshow]

Lakoff's argument takes off when he mentions metaphor.

[HalfassDIY]

Interesting that it takes a linguist to give us a satisfying understanding of this question about mathematics.

[SA-xvgc]

In Sir Roger Penrose's " Is Mathematics Invented or Discovered?" he claims that Mathematics is discovered and I find Sir Penrose's argument very hard to debate.
In essence, he reminded us that there are examples of physical theories (he mentioned the general theory of relativity) which, at the time of their formulation, gave predictions that were many orders of magnitude more precise than the accuracy of the available observations and so observation could not have had any effect on the predictions of the theory. Certainly, the available observations act as a guide in designing the theory in such a way as to agree with the already available observational data. Many years later, when the accuracy in the observations had improved sufficiently, the new, more accurate observational data turned out to verify/agree with the predictions of the theory. It is in that sense that he assigns an independent reality to mathematics which is reflected in the physical world and I find this argument extremely convincing.
Is it however safe to follow and trust the conclusions drawn by a mathematical physical theory blindly until the very end i.e. to arbitrary large energy scales way beyond the ones we have already probed with experiment or at least we have the potential to probe in the near future? Probably not, since history has revealed that theories that give accurate descriptions of reality in certain energy scales have to be replaced by more accurate theories to describe phenomena in higher energy scales. This fact however does not in my opinion invalidate the former argument about the independent reality of mathematics it only indicates that this mathematical reality is more elaborate than our currently best theories and a refinement of the latter is necessary in order to encapsulate the mathematical/physical reality.
TDP.

[loyalkeyboardcoolkid-co-le]

George Lakeoff said that "the marvelous thing about mathematics is that we can create mathematics with our brains that fit phenomena in the world remarkably." But Robert's question was " there was a time when there was no human minds but the physical world worked and that physical world seems to be described by mathematics." The question was not answered.

[APaleDot]

As Maurice Merleau-Ponty wrote:
"perception is a nascent logos"

[TheFlamingChips]

Although he beautifully explained how math was invented, I'm still not sure if he fully answered the question. At the end he says the log is in our mind, not in the spiral galaxy. But that log perfectly explains the spiral galaxy so in that sense the log IS in the galaxy, therefore math was also discovered.

[arlieferguson]

A lot of people in the comments seem to be missing that Lakoff's contention is that The relationship between mathematics and reality is not an either or relationship. The two fit together as part of one whole that the brain puts together. There is no mind independent mathematics because it requires a mind. At the same time, the universe is always already out there to be seen as mathematical and can be seen as mathematical even if we are not actually looking at anything. Mathematics is surely a construct and assistive symbols just like any language.

[Jalcolm]

The discussion (as is so often the case) is about definitions. The philosophical term is A Priori - that which has "existence" independent of experience. How we come to be aware of this realm does not necessarily define the realm itself. Plato got a bit carried away in imagining that "the forms" were truer than the world of experience, when it is better to understand the
a priori as different to, not realer than, the world of experience. Obviously, we invent the things that we discover, and vice versa.

[williamwolfe]

Cool, it took Lakoff 2 minutes and 8 seconds to say, "Metaphor". Anyway, Mathematics as Metaphor is an interesting twist.

[theophilus]

Lackoff clearly ignores the opening question. Instead, he talks about the psychological, cognitive and (to some extent) the neurophysiological and evolutionary processes that seem to be involved in acquiring our concepts of number and learning mathematics. This is very interesting and important stuff and we need not deny a word of it, but it does not show that, say, the number '3' is neither (in some sense) invented nor discovered. Lackoff seems just to confuse talk about the _nature_ of mathematics with talk of _thinking_ about mathematics as we learn it - the category mistake of all psychologisms.

[divided_and_conquered]

FINALLY!!! - At least someone out there has a brain! Lakoff is 100% accurate on his final conclusion. I'm glad to see at least someone doesn't make a religion out of the question of mathematics being invented or discovered! Finally! Take THAT, Penrose!! lol

[filosofiadetalhista]

I would like to dig deeper into Lakoff's and Nuñez's work. I do not know how metaphors work in cognitive linguistics, so I'm not even sure how they could be as precise as mathematical notions. I also do not know how metaphors could capture completed infinities and other mathematical oddities. Furthermore, his improvised answer to the "unreasonable effectiveness of mathematics in science" was unsatisfactory, but perhaps he has done better in written work. Finally, the whole discussion about mathematics being or not being in the world would become interesting only if made more precise.

My own opinion about the latter problem is this. There is a pretty clear sense in which (some) mathematics is in the world: physical objects bear relations among themselves which bear some morphism to some mathematical structures. Here is an example. The particles in helicoidal nebulae bear spatial relations which are (at least approximately) monomorphic to an ideal helix. That should suffice, should it not?

[shaunmcinnis]

11:05 "We create mathematics that fit". Couldn't agree more. Based on our limited human comprehension. IMHO, "The language of Math" is dictated by the physical laws that pre-existed and is the language we use to interpret them. If the physical laws were different, then the math may be as well.