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]
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.
From the wikipedia source:
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
And Im wodering why Im not looking more into the philosophy of math, i find this interesting because I am mainly an intuition driven person. Any ideas for courses online?
Tell me more about the difference between "potential infinity" and "actual infinity".
Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity?
I've been thinking a lot about infinity this past week, mostly because of Conway's lectures. I'm trying to figure out whether the following logic makes sense.
- Assuming that the universe is finite
-> Infinity cannot be approximated with finite numbers (MIP* = RE) [1]
-> Numbers cannot have infinite strings of digits
-> The future can never be perfectly preordained (Gisin) [2]
-> Free will exists (Conway) [3]
This is not a proof that free will exists! But I think this is a consistency proof that free will is finite.
Now you're telling me that there's another kind of infinity, and I want to know what that means for this thought.
The reason why intuitionist mathematics is useful in QM is because it can be viewed as resource based logic.
But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.
> Applying intuitionist mathematics to physics we can come to the conclusion that time flows
Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the context of classical physics.
> 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]
The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so proofs have a direct translation to computation.
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.
A fignaft is a small pink bird from the island of slamp. It has a short triangular beak well adapted to plucking seeds out of the hard cones of the humnit bush. It was nearly hunted to extinction by the umaglots, until their chief shaman declared the bird to be a repository of the souls of men lost at sea. Since then it has made a remarkable recovery.
In contrast to the mental process required to invent an imaginary bird such as this, the folks who discovered DNA used tools to image cell internals such that the structure of the contents was revealed.
When DNA was discovered, something about the structure of the external world was made clear to us, and many other processes could be clarified as a result. The fignaft bird, on the other hand, only reveals the contents of my mind, and is constructed entirely of existing knowledge.
> The question assumes that there is a difference between invention and discovery. There isn't. Invention is a kind of discovery.
this is self-contradictory. if invention is a kind of discovery, then they aren't equivalent as you say.
invention and discovery are clearly different but related concepts. hence the question. i didn't invent the sun. i discovered it, for myself, the first time i saw it. this is a structure that exists.
the mathematical question boils down to whether the structures that we find are discovered or conjured up. our physical inventions do discover new things, but these are things that existed whether the invention happened or not. the invention is simply a process.
>The apparent distinction arises from our separation of physical and mental processes.
You make the distinction based on (physical or mental) processes involved in the discovery/invention. I wouldn't care so much about how the discovery/invention came to be and look more into what the discovery/invention is about.
Usually, you can discover things which are present in the real world.
Mathematicians are rarely interested in the real world. They invent things which are true (or false) regardless of the physical world.
> The apparent distinction arises from our separation of physical and mental processes.
I'm not sure I understand this either. I think our ordinary intuitions would say that both invention and discovery are mental processes. They're just different kinds of mental processes: invention has an element of arbitrariness or choice in it that discovery does not.
I have no problem with saying that mental processes are physical processes; I just don't think that means invention is a kind of discovery.
I think a useful dichotomy is between whether an intellectual product's existence is possible without the existence of the person who created it. Copyrightable works typically could not exist without their creator. Only Prince was capable of creating When Doves Cry and only Jane Austen was capable of writing Pride and Prejudice. But inventions share with mathematical discoveries that they can (and often are) created or discovered separately by more than one person.
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.)
Presumably you would agree that not all of discovery is also invention. So doesn't that restore the original question of asking whether mathematics is an invention or discovery which is not invention?
as if this were obvious but... If you accept the possibility of random events (something deeply related to Quantum Mechanics)... There can be lots of non-computable things out there in our minds...
The words "invention" and "discovery" are also both human conceits. It seems perfectly appropriate, then, for them to distinguish two separate other human conceits.
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.
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.
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.
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.
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?"
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.)
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.
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?
Think of it this way: Anyone can invent a new song, and no matter how derivative or lacking in artistic merit, they can still claim to have composed a new piece of music.(La-la-la-laaa. There, I just did it! Short but sweet.)
But you cannot just "invent" a new proof to the Pythagorean theorem. Any proof that a^2 + b^2 = c^2 will have to clear the hurdle of being demonstrably "true."
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.
> 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?
You can only reach this conclusion if you categorize math as "not real". If everything we interact with is a perfect mathematical object, I prefer to take that as evidence that all mathematical objects may be real, however abstract they seem.
> 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.
What’s depressing about being a nanotchmological robot attempting to conquer the solar system and spread his technology and nanotechnology? It sounds rather exciting and fascinating. Perhaps if the depression is too severe you need to check out your programming ;-)
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?
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.
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!
> 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.
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?
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.
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.
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.
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.
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.
[+] [-] DeathArrow|5 years ago|reply
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
[1] https://www.quantamagazine.org/does-time-really-flow-new-clu...
[+] [-] hliyan|5 years ago|reply
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.
[+] [-] tartoran|5 years ago|reply
And Im wodering why Im not looking more into the philosophy of math, i find this interesting because I am mainly an intuition driven person. Any ideas for courses online?
[+] [-] peterburkimsher|5 years ago|reply
Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity?
I've been thinking a lot about infinity this past week, mostly because of Conway's lectures. I'm trying to figure out whether the following logic makes sense.
- Assuming that the universe is finite -> Infinity cannot be approximated with finite numbers (MIP* = RE) [1] -> Numbers cannot have infinite strings of digits -> The future can never be perfectly preordained (Gisin) [2] -> Free will exists (Conway) [3]
This is not a proof that free will exists! But I think this is a consistency proof that free will is finite.
Now you're telling me that there's another kind of infinity, and I want to know what that means for this thought.
[1] https://www.quantamagazine.org/landmark-computer-science-pro...
[2] https://www.quantamagazine.org/does-time-really-flow-new-clu...
[3] https://www.youtube.com/watch?v=tmx2tpcdKZY&feature=youtu.be...
[+] [-] mikorym|5 years ago|reply
But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.
[+] [-] hyw|5 years ago|reply
Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the context of classical physics.
[+] [-] naasking|5 years ago|reply
The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so proofs have a direct translation to computation.
[+] [-] lisper|5 years ago|reply
[+] [-] jhedwards|5 years ago|reply
A fignaft is a small pink bird from the island of slamp. It has a short triangular beak well adapted to plucking seeds out of the hard cones of the humnit bush. It was nearly hunted to extinction by the umaglots, until their chief shaman declared the bird to be a repository of the souls of men lost at sea. Since then it has made a remarkable recovery.
In contrast to the mental process required to invent an imaginary bird such as this, the folks who discovered DNA used tools to image cell internals such that the structure of the contents was revealed.
When DNA was discovered, something about the structure of the external world was made clear to us, and many other processes could be clarified as a result. The fignaft bird, on the other hand, only reveals the contents of my mind, and is constructed entirely of existing knowledge.
[+] [-] nikofeyn|5 years ago|reply
this is self-contradictory. if invention is a kind of discovery, then they aren't equivalent as you say.
invention and discovery are clearly different but related concepts. hence the question. i didn't invent the sun. i discovered it, for myself, the first time i saw it. this is a structure that exists.
the mathematical question boils down to whether the structures that we find are discovered or conjured up. our physical inventions do discover new things, but these are things that existed whether the invention happened or not. the invention is simply a process.
[+] [-] dorgo|5 years ago|reply
You make the distinction based on (physical or mental) processes involved in the discovery/invention. I wouldn't care so much about how the discovery/invention came to be and look more into what the discovery/invention is about.
Usually, you can discover things which are present in the real world. Mathematicians are rarely interested in the real world. They invent things which are true (or false) regardless of the physical world.
[+] [-] pdonis|5 years ago|reply
I'm not sure I understand this either. I think our ordinary intuitions would say that both invention and discovery are mental processes. They're just different kinds of mental processes: invention has an element of arbitrariness or choice in it that discovery does not.
I have no problem with saying that mental processes are physical processes; I just don't think that means invention is a kind of discovery.
[+] [-] toasterlovin|5 years ago|reply
[+] [-] pdonis|5 years ago|reply
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.)
[+] [-] dntbnmpls|5 years ago|reply
The distinction is that discovery is of something that exists without human intervention and invention exists because of human intervention.
For example, telescopes are invented ( human made ). The moons of Saturn are discovered ( not human made ).
You created a straw man and argued about dualism. The question of discovery and invention has nothing to do with mind/body problem.
[+] [-] ambulancechaser|5 years ago|reply
Presumably you would agree that not all of discovery is also invention. So doesn't that restore the original question of asking whether mathematics is an invention or discovery which is not invention?
[+] [-] pfortuny|5 years ago|reply
> mental processes are computational processes
as if this were obvious but... If you accept the possibility of random events (something deeply related to Quantum Mechanics)... There can be lots of non-computable things out there in our minds...
[+] [-] ip26|5 years ago|reply
[+] [-] yters|5 years ago|reply
[+] [-] tim333|5 years ago|reply
[+] [-] koheripbal|5 years ago|reply
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[+] [-] danck|5 years ago|reply
The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.
[+] [-] keiferski|5 years ago|reply
https://plato.stanford.edu/entries/philosophy-mathematics/
[+] [-] Reedx|5 years ago|reply
Max Tegmark: https://www.youtube.com/watch?v=ybIxWQKZss8
Stephen Wolfram: https://www.youtube.com/watch?v=nUCwtLTUPQ4
Steven Weinberg: 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.
[+] [-] amarte|5 years ago|reply
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.
[+] [-] DeathArrow|5 years ago|reply
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.
[+] [-] lavp|5 years ago|reply
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|5 years ago|reply
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
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|5 years ago|reply
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|5 years ago|reply
[+] [-] captain_clam|5 years ago|reply
But you cannot just "invent" a new proof to the Pythagorean theorem. Any proof that a^2 + b^2 = c^2 will have to clear the hurdle of being demonstrably "true."
[+] [-] DeathArrow|5 years ago|reply
[+] [-] itvision|5 years ago|reply
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.
[+] [-] kd5bjo|5 years ago|reply
You can only reach this conclusion if you categorize math as "not real". If everything we interact with is a perfect mathematical object, I prefer to take that as evidence that all mathematical objects may be real, however abstract they seem.
[+] [-] otabdeveloper4|5 years ago|reply
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.
[+] [-] novalis78|5 years ago|reply
[+] [-] mrtksn|5 years ago|reply
What struck me was what prof. Michael Merrifield said at the end of his explanation: 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|5 years ago|reply
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|5 years ago|reply
[+] [-] noodles_ftw|5 years ago|reply
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.
[+] [-] raghavtoshniwal|5 years ago|reply
[+] [-] magicalhippo|5 years ago|reply
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
[+] [-] deltron3030|5 years ago|reply
[+] [-] ebj73|5 years ago|reply
[+] [-] mykhamill|5 years ago|reply
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|5 years ago|reply
[+] [-] raincom|5 years ago|reply
We see similar disputes in philosophy of (natural) sciences: for instance, instrumentalism doesn't subscribe to the ontology of realism.