
Roger Penrose – Is Mathematics Invented or Discovered? [video] - mmmBacon
https://www.youtube.com/watch?v=ujvS2K06dg4
======
DeathArrow
Intuitionist mathematics claims that mathematics is purely the result of the
constructive mental activity of humans rather than the discovery of
fundamental principles existing in an objective reality. [0]

In intuitionist mathematics there is only potential infinity, no actual
infinity. Constructive set theory differs from Zermelo set theory.

That has many consequences in practice. Applying intuitionist mathematics to
physics we can come to the conclusion that time flows and it helps reconcile
quantum mechanics with general relativity.[1]

[0]
[https://en.wikipedia.org/wiki/Intuitionism](https://en.wikipedia.org/wiki/Intuitionism)

[1] [https://www.quantamagazine.org/does-time-really-flow-new-
clu...](https://www.quantamagazine.org/does-time-really-flow-new-clues-come-
from-a-century-old-approach-to-math-20200407/)

~~~
hliyan
I've always thought math as we know it is a result of both deep underlying
relationships between natural constructs _and_ how we perceive them. For
example, a species who has no sense of vision will not develop geometry the
same way we would. A species that has a sense of vision that also includes a
direct distance perception (as opposed to our stereoscopic vision) will
probably come up with a very different form of geometry.

Though I'd like to think that most species would come up with some version of
calculus, even though the notation will be obviously different. Afterall, two
of our own species did so independently.

~~~
jbotz
"two of our own species [discovered calculus]"..? Which two species are those?
I assume by "own own" you mean Terran species, and one of them is Homo
Sapiens, which is the other one? Really curious...

~~~
red75prime
Leibniz and Newton. Two [members] of our own species.

~~~
sunstone
And Archimedes makes three.

------
lisper
The question assumes that there is a difference between invention and
discovery. There isn't. Invention is a kind of discovery. The apparent
distinction arises from our separation of physical and mental processes. But
this is a purely artificial separation, a human conceit, because (unless
you're a dualist) mental processes _are_ physical processes. Specifically,
mental processes are _computational_ processes, which are physical. So it's
not surprising that the structure of the physical world should be reflected
back on itself in the structure of mental processes. It's universal Turing
machines all the way down.

~~~
pdonis
_> Invention is a kind of discovery._

Some inventions can be thought of this way, but I don't think all inventions
can. For a relevant example, human mathematical notation is a human invention
that I don't think can be usefully thought of as a kind of discovery. (Another
poster upthread mentioned Tegmark's response drawing a distinction between the
structure of mathematics, which we discover, and the language we use to
describe it, which we invent.)

~~~
dTal
I think notation can certainly be regarded as a kind of discovery. As time
goes on, notation improves dramatically as a tool of thought. This improvement
is a result of feedback over what works and what doesn't. The feedback is
discovery.

Invention = discovery in "solution space".

~~~
stjohnswarts
I don't understand why it is useful to argue "invention is discovery is
invention". They are fairly well defined in the dictionary and it's a useful
distinction to humans. Sure there is a gray area of a wide swathe of overlap,
but in general they are useful as commonly understood.

------
danck
If you only look at mathematics I think it's simply: \- Axioms are _invented_
\- Conclusions are _discovered_

The magic part for me is that some axioms have been chosen so well that their
conclusions are confirmed in the real world.

~~~
red75prime
Arithmetic existed long before its axiomatization. Arithmetic was useful and
no one stumbled upon contradictions in it. So it was natural to suppose that
it can be described by some axiomatic system. Peano found it.

~~~
kevin_thibedeau
It is a system for modeling concepts invented by man. Everything that falls
out of such a system is a product of the invention. Numbers don't inherently
exist. Everything derived from that concept can't be a "discovery".

~~~
red75prime
> Numbers don't inherently exist

How do we know that it is true?

~~~
kevin_thibedeau
They are symbols that we assign arbitrary meaning to. They are useful because
of the axiomatic framework constructed to support them.

------
keiferski
The SEP article is an excellent introduction to the philosophy of mathematics:

[https://plato.stanford.edu/entries/philosophy-
mathematics/](https://plato.stanford.edu/entries/philosophy-mathematics/)

------
Reedx
Same question also answered by

Max Tegmark:
[https://www.youtube.com/watch?v=ybIxWQKZss8](https://www.youtube.com/watch?v=ybIxWQKZss8)

Stephen Wolfram:
[https://www.youtube.com/watch?v=nUCwtLTUPQ4](https://www.youtube.com/watch?v=nUCwtLTUPQ4)

Steven Weinberg:
[https://www.youtube.com/watch?v=NpMk9G-ddiM](https://www.youtube.com/watch?v=NpMk9G-ddiM)

I particularly liked Max's answer who neatly makes the distinction between the
structure (we discover) and the language (we invent). We're free to invent the
names, but not the structures.

~~~
7thaccount
Max Tegmark has some fun books. Apparently he got to be at the forefront of
some discoveries as he and his wife knew C++ and would do all the after the
fact data analysis.

If you've ever been around a campfire with a good storyteller that has their
audience at the edge of their seat, I felt like this in some spots.

------
amarte
Imagine an interview with a great painter. The painter is asked about the
nature of painting and he responds that when he sets out to work in his studio
and his brush strokes a canvass, he is discovering the fundamental nature of
reality. He doubles down and exclaims that, in fact, reality is actually JUST
lines and curves and shades and colors, and his proof is, well, look at how
accurate his paintings are! Let's pretend that he actually is a very skilled
painter, and many critics have marveled at the extent to which his paintings
are indistinguishable from his subject matter. Still though, the interviewer
clears his throat after an awkward pause and continues on to the next
question.

To me, math is a medium of description much like painting or writing. It's
units are not colors or words but points. A point, or "that which has no
part", is much finer and carries with it far less baggage than something like
a word. Points can be assigned numerical values and played with in clever
ways. You can even sprinkle a fine dust of them over anything observable and
create a copy of it to arbitrary degrees of precision.

I know math is far more than just the study of points, but I'm not convinced
that math is anything more than our capacity to distinguish and describe
extended to its limit. I also don't mean to belittle the accomplishments of
mathematicians and theoreticians, I just think it's more reasonable to say
that math is JUST the limit of description than it is to say that reality is
JUST math.

After reading through some of the comments, I think many of us are on the same
page.

~~~
bloomberg2020
Agreed

The crux of it is obvious: math is true from the perspective of a human
consciousness

It’s defined after cognitive measure.

I won’t say it’s whittled down.

I would say ... it’s sight, sound, touch... attempt at quantifying our
qualified experience

All the best math nerds were often deeply gifted and practiced forms of human
art: poetry, music, sculpture... and many artists good at math, Brian May, for
one example

IMO to get math one needs to build all those abilities

It’s really the people I know that don’t do that see math as a boring toolbox
of rules and points

Not saying you’re not that

Math is not a thing like a rock because our ability to behave like more than a
rock gave us the ability to define math. It has to be more than a simple this
or that

------
DeathArrow
I am not sure of the implications. Does it even matter if Math is invented or
discovered? Maybe it's both, and it isn't a contradiction between invention
and discovering?

We describe things having a certain radiation wavelength as having the color
yellow. In that sense, yellow it's an invention. That doesn't mean the
radiation doesn't exist.

But to complicate things a bit, some things don't exist unless we observe
them. This is the case with states of particles described by quantum
mechanics.

Math is more than a science, is the sciences upon which most other sciences
and tech are founded. You can model anything in a computer and running on a
computer using math. You can describe logic, natural language, technology,
biology using math.

In that sense, being the building block of other sciences, math is more akin
to a language. Two physicist use math in almost the same way two people use
English to describe things and communicate ideas.

But math has building blocks, too. Set of axioms upon which any mathematical
object and theory can be constructed. The most popular as of now is Zermelo
set theory. There are more such fundamental theories, sometimes very different
between them.

So, to see if Math is discovered or invented, the easy thing to do is to see
if a set of axioms can be discovered or is invented.

~~~
goatlover
> Does it even matter if Math is invented or discovered?

It matters for philosophical inquiry. Does philosophy matter? It matters for
people who find it valuable, the same way art, music or literature matters.

~~~
guerrilla
You think logic, the art of making distinctions, the study of critical
thinking and the foundation of mathematics only matter in the same way art
does?

~~~
inopinatus
I’d say they don’t matter nearly so much as art. The most cosmically
significant export of humanity to date is Voyager’s golden record.

~~~
guerrilla
Good thing the founders of modern science, mathematics, ethics and political
theory didn't share the same opinion.

~~~
inopinatus
Ridiculous hubris, to claim to speak on their behalf, to claim to know the
opinion of so many and diverse minds, and in such a crude misdirection to
boot.

~~~
guerrilla
Good thing they wrote volumes about it which have debated and summarized to
the hilt.

~~~
inopinatus
The summary is that without art, humans are cosmically uninteresting.

------
lavp
I find mathematics to be the most beautiful thing in the universe, because it
is the only thing I can truly think of as being perfect in every way. Existing
only as an abstract concept, it still manages to find its way into every
single aspect of our lives.

Two completely remote civilizations will still have the same mathematics. Sure
one may have a more developed understanding, but if both civs wondered about
how to get the hypotenuse of a right-angled triangle, they would both end up
with Pythagoras' theorem. The only thing invented in math is our language and
representation of such concepts.

Many people believe that mathematics becomes invented the further up you go
like pure mathematics and I can understand why this perspective would come
through, but take the example of G.H. Hardy who famously said “The Theory of
Numbers has always been regarded as one of the most obviously useless branches
of Pure Mathematics”. It could be argued at the time that these branches were
simply invented mathematics because there was no practical application of it,
but 30 years following his death came breakthroughs in cryptography. All of a
sudden, this was no longer an invention. Calculus is not seen as an invention
but a discovery because of all of its insane number of applications in physics
and other areas.

Many areas of math that have yet to find practical use will always come under
the scrutiny of being 'invented' but that simply isn't the case.

I firmly believe that all areas of mathematics are practical to the universe
and its wonders (and therefore discovered), whether or not we can achieve a
level to use such mathematics (or experience it) is a different matter.

I believe that at its core, mathematics is the universe. To say that we have
invented it is arrogant and completely strives it of its beauty.

------
Wistar
Dr. Hannah Fry's 3-part BBC documentary series, "Magic Numbers: Hannah Fry's
Mysterious World of Maths," explores this very question in the first episode,
"Numbers As God."

The episode description reads: "Documentary series in which Dr Hannah Fry
explores the mystery of maths. Is it invented like a language, or is it
discovered and part of the fabric of the universe?"

[https://www.bbc.co.uk/programmes/b0bn6wtp](https://www.bbc.co.uk/programmes/b0bn6wtp)

She interviews a number of prominent mathematicians and scientists, such as
Brian Greene, and they certainly don't agree one way or the other in the
invented/discovered question.

(Alas, I now see that the series is listed as unavailable on the BBC site but
I watched it, I _think_ on Amazon Prime or maybe youtube.)

------
justforfunhere
Could the same be said about just about everything?

Like Music. Is a Song just a combination of notes, beats, intervals, voice
etc. waiting to be discovered? Or is it something that a musician invents in
her brain through talent, experience, practice and trail and error.

Or for that matter, a startup idea? A product/service that would bring immense
value to its consumers, but it is not there yet, waiting to be discovered.

I guess philosophers must have dwelled on such questions before.

~~~
cpsempek
I think I get your point and I‘m open to ideas which drive home your point in
a more precise way. However, the main difference between concerning ourselves
with whether or not mathematical objects exists vs music or start-up ideas is
their place in building a foundation for understanding reality. Science has
been a productive program for understanding reality. It happens to be
underpinned by math. And as a result, we have encountered some pretty
interesting relationships between what the math tells us and what we observe.
For instance, neutrinos and black holes were known to exist mathematically
before they were ever observed and measured “in reality”. It’s not clear to me
that music and start-up ideas hold up to math when it comes to being a tool
for understanding or describing our reality, or perception the rig. Math is
somehow fundamental to many successful enterprises which seek to describe and
explain reality. Therefore, it becomes the point of focus of much
philosophical inquiry when seeking to understand reality. Hence the
foundational question, do mathematical objects exist? And not, does music or
blockchain kittens exist?

~~~
kruasan
>do mathematical objects exist?

Yes, they do, in an abstract sense. Existence in math means a very different
thing from existence in physics. There are mathematical objects that exist,
and those that cannot and do not exist. Someone posted a SEP entry earlier,
which is a good start on this topic.

>the main difference between concerning ourselves with whether or not
mathematical objects exists vs music

There is no 'vs', because there is no difference. Music _is_ math, any song or
sound is a mathematical object. A physical waveform that you hear can be
encoded digitally in numbers: ones and zeros. So any given wav/flac file is
just a bunch of numbers that give rise to the qualitative experience of sound,
when interpreted in a certain way. For example, a digital waveform consists of
samples, each sample takes 16 bits to encode. Sampling rate of 44.1kHz is
44,100 samples per second. So you have 16 bits per sample x 44100 samples per
second per channel x 2 channels x 300 seconds = 2^423,360,000 possible
permutations of a 5-minute audio file without compression, which is a number
with over 127 million digits. A little percentage of these permutations would
count as music (even if your tastes are really diversified), most of it would
just be noise. But all these possible 5-minute audio files include not only
every song and every performance that existed or will exist. They also include
every possible sound recording: songs that will not be written, Paul Graham
saying that he hates HN, Paul Graham saying that he loves me and the rest of
the file is silence, you and me discussing this topic with Plato for 5
minutes, etc, etc. The data exists and can be discovered and listened to, even
though some of these examples are obviously not physically possible (i.e.
Plato is dead).

So all music already exists mathematically, and it can be a useful mindset
that your job is to _discover it_. A lot of musicians see it that way, Tessa
Violet, for example:
[https://youtu.be/QzBoGVToWEo?t=342](https://youtu.be/QzBoGVToWEo?t=342)

------
itvision
There's one thing I cannot wrap my head around.

Math is purely abstract science yet it describes the world around us to the
utmost precision. Does that mean that this world is simply ... a math model
which means everything around us is ... not real?

I've long stopped believing in free will because everything points at it being
an illusion of our brain because we're a product of this world and we had no
chance of influencing the conditions which brought us to life, and even after
our minds and consciousnesses form it's hard to believe they are fully
autonomous and not simply a function of the processess in our brains we're
simply not aware of.

If you think about all of it, it becomes utterly depressing as you begin to
realize you're a biological robot, a byproduct of the universe evolution which
couldn't care less about our species and this little tiny blue planet.

~~~
otabdeveloper4
> Math ... describes the world around us to the utmost precision.

It does no such thing. The models we create using mathematical language do.

Math is a precise language that can be used to describe the world around us
precisely. It can also be used to describe utter nonsense or utter fantasy
precisely.

~~~
Tainnor
Yep, the counterpoint to the unreasonable effectiveness of maths in the
natural sciences is accompanied by the unreasonable ineffectiveness of maths
in the social sciences, humanities, etc. Although at least to some extent
statistics is the one branch of mathematics that does seem to be applicable
(but the way this influences results, as opposed to methods, seems to be
mathematically not that interesting).

There have been a lot of efforts of modelling social phenomena, the arts, etc.
mathematically and while there are some interesting partial results (e.g. that
languages can to some reasonable extent be analysed by parse trees or some
aspects of music), most "grander" theories I have seen do not really stand up
to much scrutiny.

I'm not an expert, but I could assume that part of this is due to a lot of
nonlinearity in the phenomena studies, which means that many classical methods
don't work well; maybe chaos theory etc. could shed more light on these
things, but I don't know enough about it.

~~~
naasking
> Yep, the counterpoint to the unreasonable effectiveness of maths in the
> natural sciences is accompanied by the unreasonable ineffectiveness of maths
> in the social sciences, humanities, etc.

Except it's not ineffective at all in these subjects. Social science
experiments have so many variables that the small experiments that can be
conducted given the financial resources are insufficient to infer a good
model. This is not a math problem, it's a money problem.

~~~
Tainnor
If it's ineffective in practice, I consider it to be ineffective.

But even if you could design an experiment perfectly and come up with some
strong statistical evidence and then had other means to tease out what is
actually causal and what is only correlational, you'd only know what
influences what and not necessarily why. But yes, I did say that
statistics/probability is some rare exception.

You can study group theory and understand the way physical forces work better,
or hilbert spaces to understand quantum mechanics, but I haven't yet heard of
anyone who has studied topology or galois theory and found that incredibly
useful for understanding social phenomena better.

~~~
naasking
> If it's ineffective in practice, I consider it to be ineffective.

Except you have no way to conclude that it's ineffective. Analytical solutions
require data to study. Without data, or with little data, what analysis are
you going to perform? At best, broad statistical correlations, which is
exactly what we find.

You're effectively claiming that spoons are ineffective at a restaurant that
provides only forks. Well no, if a spoon were available, then it would
probably work just fine.

> but I haven't yet heard of anyone who has studied topology or galois theory
> and found that incredibly useful for understanding social phenomena better.

After 5 minutes of Googling:

* Power laws: [https://en.wikipedia.org/wiki/Power_law#General_science](https://en.wikipedia.org/wiki/Power_law#General_science)

* Network theory: [https://en.wikipedia.org/wiki/Network_theory](https://en.wikipedia.org/wiki/Network_theory)

* There's a journal specifically for mathematical social sciences: [https://www.journals.elsevier.com/mathematical-social-scienc...](https://www.journals.elsevier.com/mathematical-social-sciences/)

* Economics, game theory, and social choice theory are all examples employing heavy analytical problem solving to social problems

* Quantum mechanics applied to social sciences: [https://www.cambridge.org/core/books/quantum-social-science/...](https://www.cambridge.org/core/books/quantum-social-science/156E054488073E842C6A3C598FC8DC87#fndtn-information)

The main stumbling block is that mathematicians are interested in mathematical
problems, and so they make a common but mistaken assumption that social
sciences either don't have such problems, or they are too messy for elegant
math. Take it from a mathematician, this is incorrect:
[https://www.mathtube.org/sites/default/files/lecture-
notes/S...](https://www.mathtube.org/sites/default/files/lecture-
notes/Saari.pdf)

~~~
Tainnor
You may have a point. And I did forget about things like social choice theory
or game theory (although I'd assume that partially this is also due to e.g.
social choice procedures or "games" often being very limited and artificial
settings where by their very nature the relevant space of options/outcomes can
be explored in some systematic way, which is generally less the case in more
organic, complex settings, such as e.g. gradual societal changes).

When it comes to economics, I know that a lot of people don't agree with the
basis of many mathematical models that are used, but I'm not an economist, so
I can't speak to that.

So maybe I was overzealous in discounting mathematics for the social sciences
altogether. Still, I would contend (albeit with much less evidence):

\- There's some measure of people trying to construe "nice models" of things
in those subjects instead of trying to make sure they agree with reality. I
have a degree in linguistics and I've seen this over and over. Most of these
models haven't convinced me at all.

\- The amount of maths that you either need or at least benefit from in order
to do good research in such areas is still substantially lower than in, say,
physics. I think it's still to be noted how we can describe much of physics
just with a small number of economic and elegant models. I haven't seen
anything comparable in any of the social sciences.

------
mrtksn
Recently I watched Sixty Symbols video on why light is slower in glass and
there are multiple explanations on it.

What struck me was what prof. Michael Merrifield said at the end of his
explanation:
[https://www.youtube.com/watch?v=CiHN0ZWE5bk](https://www.youtube.com/watch?v=CiHN0ZWE5bk)

On the question about what is the reality, what explanation is the true one,
Merrifield said that the Math works on all of the explanations and what those
explanations do is simply to model the behaviour of nature and not necessarily
reflect the reality.

That's how Newtonian physics and relativistic physics are both correct models,
tools to model nature and simply can be used to whenever suitable.

Wouldn't that mean that mathematics is just an invented tool to reason about
physical models?

~~~
iNic
Well yes and no. A lot of mathematics found its roots in trying to model the
real world, however I would then argue that the truths we prove about these
models are truly discoveries. And with these discoveries we can often
generalize the setting to more abstract formulations, independent of the
physical reality.

I think that this can be seen in the fact that sometimes mathematicians and
especially physicists can reason about objects that they are not sure about
what the right definition should be. Many mathematicians reasoned about
continuous functions long before we had a concrete definition of them. But
when Cauchy introduced the definition and Weierstrass proved it is equivalent
to preserving limits (which was the intuition at the time), we had not truly
discovered something new and mathematical.

This whole idea was then generalized to topology when it was shown that pre
images of continuous functions preserve the "openess" of sets, i.e. we
realized that no concept of distance was not needed to describe continuity,
which is very surprising.

------
guerrilla
I'll never understand how someone otherwise so apparently intelligent can be
so religious. Weird, the systems that we spent massive amounts of energy
designing to precisely describe reality do that better than all the ones that
we threw out along the way!

~~~
noodles_ftw
> I'll never understand how someone otherwise so apparently intelligent can be
> so religious.

Why not? Religious people believe that God is the Most Wise. Mathematics in
nature attest to that attribute of God (as well as to other attributes of
Him). An intelligent and religious person would recognize that, knowing it is
God who came up with all the rules that keep the universe in balance.

I'll never understand why some people believe science and creationism can't go
hand in hand.

~~~
jl6
Careful with your use of the word creationism. I guess you mean it in the
sense of the very vague and general claim that “god created the universe”, but
in my experience it is used to refer to a much more specific set of claims,
such as “the Earth is about 6000 years old”, which is absolutely not
compatible with science.

~~~
noodles_ftw
According to Wikipedia: 'Creationism is the religious belief that nature, and
aspects such as the universe, Earth, life, and humans, originated with
supernatural acts of divine creation.'.

I choose my words carefully. I get what you're saying though, but I think
that's more of a subjective matter.

------
raghavtoshniwal
Last week I came across another fascinating podcast by Penrose that you folks
might also like:
[https://www.youtube.com/watch?v=orMtwOz6Db0](https://www.youtube.com/watch?v=orMtwOz6Db0)

~~~
peterburkimsher
Thanks for the link. At 1:02:57, Penrose proposes "infinite cycles of the big
bang"

[https://youtu.be/orMtwOz6Db0?t=3777](https://youtu.be/orMtwOz6Db0?t=3777)

Is that multiverse theory disproven by Wolfram's latest blog post?

"could there be other universes? The answer in our setup is basically no."

[https://writings.stephenwolfram.com/2020/04/finally-we-
may-h...](https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-
path-to-the-fundamental-theory-of-physics-and-its-beautiful/)

------
magicalhippo
If B follows from A via logic, but A is invented, isn't B invented as well
then?

Just to take a recent example which was mentioned here, Geometric Algebra[1].
There you assume you have some objects which aren't numbers but which when
squared equals a given number. By doing that a bunch of nice results have been
discovered.

However to me the basic premise, take some objects which aren't numbers but
which square to a number, feels very much like an invention. So as such
wouldn't the nice results be inventions as well?

[1]: [https://bivector.net/doc.html](https://bivector.net/doc.html)

~~~
analbumcover
It sounds to me like you are giving special precedence to "numbers" in your
considerations, as well as drawing your intuition for "square" from working
with the integers or the reals. A square is just the result of a binary
(product) operation where both of the operands are equal. The operation can be
defined on an algebraic object with much richer structure than the integers or
reals, and the operation itself can be much more complex than, say, integer
multiplication. So the geometric product of geometric algebra is just one
animal in the zoo of examples. You might call the specifics of the geometric
product operation an "invention," but not because the square of a non-number
can be a number.

~~~
magicalhippo
Well if one would consider numbers and their product "discovered", then my
point was that introducing the objects which have the property that they are
not numbers but square to numbers seems to me like an invention.

At least if there is to be any meaningful distinction between a discovery and
an invention.

------
deltron3030
Mathematics is just a language and likely a human discovery method for rules
that encapsulate the universe and cascade down. A method invented to discover
discoverable rules.

~~~
roywiggins
You can build all sorts of bizarre structures in ZFC that seem to have no
relationship with reality at all. You can prove that almost all real numbers
can't be computed or even described. There is an entire tower of infinities
bigger than the natural numbers that telescope up and up. The axiom of choice
gives you all sorts of nonphysical results.

The world of ZFC doesn't feel very constrained by the actual universe, is all.

And if you don't like ZFC, you can pick another set of axioms entirely- there
are several alternatives.

~~~
deltron3030
Sure, math/ZFC can describe potential and impossible universes including ours.
But the motiviation for its invention as a language was very likely to
understand relationships in our reality, physics, trade etc.

------
ebj73
A vaguely, but tangentially related fact, is that Stephen King actually
considers the stories in his books as discovered, rather than invented. He
talks about it in his book 'On Writing'. He considers the stories to be sort
of preexisting things, and his job as a writer to unearth and discover them,
rather than invent them. It's about his frame of mind while writing, I guess,
but still.

~~~
RashadSaleh
I doubt that he actually said that but I could be wrong.

~~~
Trasmatta
This is actually a big plot point in The Dark Tower books.

------
mykhamill
What can be imagined is what can be discovered.

The etymology of invent has the terms 'contrived' and 'discover' baked in. if
we take the contrived root, rather than just dead-ending to say that invent is
synonymous with discover, we find that it is rooted in the ability to
'compare' and 'imagine'. From this we can then formulate the opening
statement.

------
Koshkin
My take on this is that Mathematics is such a vast area of investigation -
much larger than Nature itself - that there is in fact a mix of both. In that
regard it is close to Engineering, where in an attempt to design something on
might stumble upon things or perform some types of research and discovery, and
vice versa.

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raincom
It is a dispute in ontology. Do mathematical objects (say, numbers, sets)
exist in the world? Some say, yes; others say, no; some others say, they exist
in another world--called Platonic world.

We see similar disputes in philosophy of (natural) sciences: for instance,
instrumentalism doesn't subscribe to the ontology of realism.

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BtM909
I read this book about it: [https://www.amazon.com/God-Mathematician-Mario-
Livio/dp/0743...](https://www.amazon.com/God-Mathematician-Mario-
Livio/dp/074329405X). Won't spoil the answer, but was nice to go through the
evolution of Mathematics.

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sunstone
Put me into the 'discovered' camp.

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bane
In most fields, you have to invent tools that enable you to discover things.
My guess is that this is also true in mathematics. We get confused sometimes
because the tools often happen to look like the discoveries because of all the
fancy greek symbols.

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ukj
The answer is: "both".

Where do Mathematicians look to discover Mathematics? In the depths of their
own minds.

The part where you "look and think deeply" is discovery. The part where you
"express your discovery in a coherent language" is invention.

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NautilusWave
I initially read the title as "Is Mathematics haunted or discovered".

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hyperpallium
Operational theory of mathematics: you define (invent) some rules, of starting
points and ways to change them. Like the rules of a game.

In a sense, all possible outcomes already exist (only to be discovered).

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scrozart
Phenomena are discovered. Mathematics are invented to describe them.

~~~
Noughmad
Not all mathematics though.

Math works by starting with a set of axioms. These are chosen, usually but not
necessarily such that they will describe some kind of physical reality. That
is the part which you invent to describe phenomena, like the Peano axioms
which describe quantities of countable things.

But then, you discover (and of course also prove) things that follow from
these axioms. So a mathematician would find that if you take this set of
axioms, then this statement is true. Is that an invention or discovery? I
would call it a discovery, although different from a discovery of a natural
phenomenon.

~~~
roywiggins
Geometry and arithmetic and calculus all existed before axioms and worked
okay. You can do quite a lot of mathematics "naively". It's only in the 20th
century that axiomatization became really important, because naive set theory
turned out not to work very well.

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theaeolist
Like in any game, you invent the rules (axioms and logical systems) then you
discover the consequences (theorems), just like you invent a maze then you
discover the way out.

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novalis78
The platonic universe of the entirety of math sounds (this week at least) like
Wolfram’s “Universe of Computational reality” of which the physical reality is
just a subset.

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xtf
Invented to describe what's being discovered.

~~~
Koshkin
You have to add: discovered _inside itself_.

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Koshkin
The linguistic conundrum here arises due to the recently adopted mindless use
of the word 'invented' instead of 'created' or 'designed.' (Indeed, Linus
_created_ Linux rather than 'invented' it as many would say today.) Invention
is a form of discovery. So, the question should be, "is mathematics created or
discovered?"

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monadic2
This is a silly semantics game and a boring one at that. the presumption that
the words are contradictory, or that mathematics is a monolithic entity that
plays a single role of invention or discovery, is a poor way of exploring the
social and cultural (co)evolution of mathematics and methodical
experimentation and observation.

~~~
Koshkin
You can't just say this to people who keep asking this question, and have been
for ages.

~~~
monadic2
Why not? It’s not like they have any obligation to respond, and someone needs
to point out the bullshit smell wafting through the room.

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naringas
A bit both:

e.g. The circle is an invention. But the number pi (it's value) is discovered

~~~
nybble41
What do you mean when you say that the circle is an invention rather than a
discovery? Are you drawing a distinction between the mathematical ideal and
the arbitrarily close approximations which can be found in nature? Though by
that criteria that case pi must also be an invention…

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jeffdavis
The expression is invented but the underlying relationships are discovered.

It's silly to think that the sqaure root of negative one is something real to
be discovered. It's just a stand-in to express complex relationships more
succinctly. The same for negative numbers, for that matter.

It would be equally silly to think that the underlying relationships are
invented. Nobody invented prime numbers, they were discovered.

~~~
Y_Y
It's just as silly to think of the "real numbers" as being discovered. They
don't exist in the real world, after all.

~~~
mikorym
They are necessary for the continuum.

If you throw away real numbers, then you lose major things in physics. I think
for example you lose the wave function in QM.

It is not a question of whether they exist in nature, but rather whether they
are the more superior technique, or not, to explain nature.

~~~
Y_Y
Wavefunctions are in fact generally complex-valued. I don't know if that
supports your argument or not.

~~~
mikorym
Complex numbers have the same cardinality as the reals.

~~~
Y_Y
That's true, but irrelevant.

Complex numbers aren't "necessary for the continuum" as you put it, and some
realists might argue that they don't hold the same "discoverability" as the
reals.

I wouldn't. I think the reals and the complex numbers have the same "realness"
and that neither represents any innate property of the physical world, despite
how obviously useful they are in physical models.

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simplesleeper
This is just the age old question of nominalism vs realism

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jelliclesfarm
Of course..we invented the ‘language of mathematics’.

I am on a Penrose kick right now. Just downloaded The Emperor’s New Mind
Audiobook.

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zeepzeep
Java was invented, Agda was discovered.

(Obviously I'm in the religion where math does exist.)

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gingahbread31
Thank you for sharing this, this an amazing interview series !

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rezeroed
Fallacy of equivocation.

