The Question : is mathematics discovered or invented?

[Hurkyl]

I imagine, if we had an operational definition of the verbs 'to invent' and 'to discover', there wouldn't be any debate over the answer to the titular question.

If we allow everyone to color those words with their own personal biases, we have no hope of getting anywhere!

 

[oldman]

I imagine, if we had an operational definition of the verbs 'to invent' and 'to discover', there wouldn't be any debate over the answer to the titular question.

If we allow everyone to color those words with their own personal biases, we have no hope of getting anywhere!

I agree. And we do have definitions: my Oxford dictionary defines invent as: Create by thought, originate, concoct. And Discover as: Find out that, become aware that. OK by me, and I'll try to use these definitions.

 

[oldman]

For me, your kind, detailed and thorough analysis of the positions taken by Mazur and Penrose (equivocation, leaning toward "Mathematics is discovered or become aware of"?) seems marred by your choice of definitions, which were:

- For the natural world, discovered means that which existed at all times.
- Invented means that which came into existence only because of happenstance.

But, to my delight this morning I found out that there are six new ducklings down at my dam. They're very much part of the natural world and weren't there yesterday. Does this mean that I didn't discover, or become aware of them? I suspect that they came into existence only because of a happenstance encounter of their mother with a drake. So must they have been invented? See the trouble one can have with such definitions!

Perhaps a little editing of your post in this respect would help me to answer it more coherently.

One remark: even eminent mathematicians like Penrose and Mazur are sometimes given to special pleading --- they love their subject so --- as we all love activities we excel at. It may be prudent to take their elevation of the nature of mathematics into an eternal truth with a pinch of salt.

 

[CaptainQuasar]

I agree that there are "real things" that behave like fields. But you are missing the distinction between a description (field) and the thing being described (an interaction). Descriptions are always invented. Things ]may be discovered (Reflect on what we are --- nothing but a chattering species of African ape, driven to describe, and describe, and describe!).

But then, shouldn't you be willing to assert that every branch of science is also invented? Because a science never is what it describes. (Okay, we might have some fun talking about linguistics, but at least things like biology and chemistry and physics.)

In the past, a common way of describing physical interactions (like gravity) was to invoke the concept of "action at a distance". This is a not-very-clear invented description of interactions, which proved to be far less practical than the "field" concept which was invented to replace it. In fact it is no longer quite fashionable to talk simply of a "gravitational field" --- describing the metric "field" of spacetime with the help of the Riemann tensor seems more accurate.

Yet the mechanism/process by which mass/energy distorts spacetime is still quite unknown. It isn't with a pair of pliers. We don't even know if there is such a mechanism/process, and just accept the distortion as a given. This embarrassment is not quite swept under the carpet, but does seem to be ignored in polite general relativistic circles, perhaps because we are still too ignorant for this level to be usefully discussed.

Come now - you aren't marshaling the argument I think you are, are you? Simply because something isn't understood that doesn't mean it's invented! In fact that seems oxymoronic - how could one not understand something that one has fully invented?

[EDIT] Re-reading I see that you did say that gravity isn't invented.

"The 'trick' of nature that is ultimately responsible for the evolution (here I go again) of the physical universe --- from a (postulated) hot plasma into the galaxies, stars, planets and other debris we have discovered." Perhaps you can say something similar in a less clumsy way?

Alas, I might have some skill at saying things more specifically or more distinctively but they usually come out clumsier and less elegant.

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On the whole I think that it's a bit of a dodge (of the original question) to say that mathematics is invented in that it involves descriptions and other intermediate representations of the thing it is studying. (I'm speaking in general, not ascribing any malice to you.) The real question, I think, is whether it's invented in the same way that something like human culture is invented - “of whole cloth,” as it were.

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[JoeDawg]

A real problem I have with people referencing Plato's forms with regards to mathematics is that Plato believed his forms existed in 'another realm', a truer realm and that the universe was simply a shadow or manifestation of this truer reality. He wasn't talking about 'the mind' or some noumenal existence. (In my opinion, the elephant in this mathematical room is the 'mind of god') And yet we have no qualitative way of referring to this 'realm of math'. I think this is backwards thinking.

This is why I would say math is an invention. Math is a human description. Ideas abstracted from the observed. It is not some thing we found, full formed, on the side of the road.

Yes there are relationships present 'in the world', but that is not math.

The Pythagorean theorem works quite well, within a certain kind of space.
But even such concepts as 1 and 2, get fuzzy in relation to things like quantum entanglement. Its not that these concepts aren't useful in their proper place, nor that they exist in some nether realm, its that they are incomplete descriptions.

This is not to say that mathematics is not complex, and has no predictive value. But its predictive value is based on accepted axioms and in many cases is no more true than a scifi action novel. Its all about human imagination. My impression is that some pretentious mathematicians would find that sort of comparison embarrassing.

 

[Q_Goest]

Regarding the definitions, you won’t find Penrose nor Mazur try to define these terms. As Hurkyl mentions, and as yourself and CaptainQ continue to prove, it is the definitions that are eluding everyone and why it is so difficult to make sense of The Question. The question is more than just the use of a few words which propose a quandry. So when I say: “I think these definitions will better coincide with what Mazure, Penrose and others who’ve written on this topic want.” I also mean that I am assuming that the authors had something other in mind that what is normally conveyed by the conventional meaning of the terms “discover” and “invent”.

Clearly, we would all agree that it’s perfectly understandable when a newspaper talks about microbes being discovered on Mars or ducklings being discovered on a pond. I have no problem with those usages. However, Penrose puts out his figure and describes it without even using the terms discover or invent. Similarly, the U of Oregon doesn’t have those words in its description. Mazur thought he was being simple and clear by discovering that he could invent a way to use the terms in his paper, but here we are scratching our heads asking what these terms mean.

One way of getting rid of the problem is to try and define what the terms mean. Another way is to restate the question in a different way. What the authors want has to be taken in context.

 

[Q_Goest]

But then, shouldn't you be willing to assert that every branch of science is also invented? Because a science never is what it describes. (Okay, we might have some fun talking about linguistics, but at least things like biology and chemistry and physics.)

I agree with oldman on this. The concept of gravity and electrical fields being some kind of three dimensional ‘thing’ is invented and is part of our human description of reality. I’d also say that all our descriptions of reality are invented, not discovered.

I’d agree with JoeDawg when he says:

The Pythagorean theorem works quite well, within a certain kind of space.
But even such concepts as 1 and 2, get fuzzy in relation to things like quantum entanglement. Its not that these concepts aren't useful in their proper place, nor that they exist in some nether realm, its that they are incomplete descriptions.

Penrose says something almost identical to this.

Similarly, all our scientific descriptions are invented because they are inexact.

 

[CaptainQuasar]

This is why I would say math is an invention. Math is a human description. Ideas abstracted from the observed. It is not some thing we found, full formed, on the side of the road.

Yes there are relationships present 'in the world', but that is not math.

Okay then, if that's not math, what is it? What is math describing?

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[CaptainQuasar]

I agree with oldman on this. The concept of gravity and electrical fields being some kind of three dimensional ‘thing’ is invented and is part of our human description of reality. I’d also say that all our descriptions of reality are invented, not discovered.

Of course descriptions are invented, they're things exclusively used by humans. That's practically a tautology. It's like saying that tools are invented or inventions are invented.

I really don't think any of this talk about descriptions being invented is at all addressing the question. It seems rather like eyebrow-arching sophistry to me.

The description of gravity is at least pointing to something more external to humanity than the description of Bilbo Baggins or the description of slapstick comedy is, right?

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[Q_Goest]

This is why I would say math is an invention. Math is a human description. Ideas abstracted from the observed.

Hi JoeDawg,
What do you think of Penrose's suggestion:

... only a small part of the world of mathematics need have relevance to the workings of the physical world.

Clearly, math isn't only abstracted from what is observed in the physical world.

The Arabic number system is a base 10 language used to describe these seemingly superfluous mathematical concepts. Similarly, we can use a binary system or we could even use the Roman Numeral system which would make the math tremendously more cumbersome. I can't imagine trying to do my job as an engineer using the Roman numeral system, but I suppose it would be possible.

The point is, these are all different languages which describe the same thing, so I'm not sure we can say it is "invented" since others in distant reaches of the universe, could similarly come up with different mathematical languages which would describe the same mathematical concepts we have which, as Penrose notes, have no relevance to the workings of the physical world.

 

[CaptainQuasar]

The point is, these are all different languages which describe the same thing, so I'm not sure we can say it is "invented" since others in distant reaches of the universe, could similarly come up with different mathematical languages which would describe the same mathematical concepts we have which, as Penrose notes, have no relevance to the workings of the physical world.

That's simply like pointing out that “chicken” is a different word in Chinese than it is in English. That doesn't mean that chickens don't exist or that avian biology is a human invention.

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[Q_Goest]

The description of gravity is at least pointing to something more external to humanity than the description of Bilbo Baggins or the description of slapstick comedy is, right?

Yes, I agree there is a something real we are referring to when we talk about gravity. Could we describe it in some other way? I think we could. All I'm suggesting is that the descriptions are invented. I agree the physical interaction itself is part of the "physical world" that is being referred to by Penrose and Mazur.

If we're to distinguish between what is invented and what is discovered, we need to also make the distinction between our descriptions of reality and reality itself.

 

[Q_Goest]

That's simply like pointing out that “chicken” is a different word in Chinese than it is in English. That doesn't mean that chickens don't exist or that avian biology is a human invention.

Not sure what you're getting at. The Chinese person is referring to something and I'm referring to something. To extend that and say now that something is invented rather than discovered is a strawman agrument.

 

[CaptainQuasar]

Yes, I agree there is a something real we are referring to when we talk about gravity. Could we describe it in some other way? I think we could. All I'm suggesting is that the descriptions are invented. I agree the physical interaction itself is part of the "physical world" that is being referred to by Penrose and Mazur.

If we're to distinguish between what is invented and what is discovered, we need to also make the distinction between our descriptions of reality and reality itself.

Penrose and Mazur are starting to talk about the physical world, eh? Very avant-garde of them.

Yes, we could definitely describe things in a different way. This is partially what I mean by saying that numbers are more of a human construct and geometric ratios are more fundamental or more real and can probably be used to describe all the same things. It's like the way that a formal grammar and a finite automaton can be isomorphic or the way many topics in set theory and graph theory are isomorphic.

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[CaptainQuasar]

Not sure what you're getting at. The Chinese person is referring to something and I'm referring to something. To extend that and say now that something is invented rather than discovered is a strawman agrument.

You're not sure what I'm saying but it's a strawman?

I may have misunderstood you and you may have already been saying the same thing I was with my chicken example. I was saying that simply because math could be formulated a different way doesn't make it an invention.

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[Q_Goest]

I may have misunderstood you and you may have already been saying the same thing I was with my chicken example. I was saying that simply because math could be formulated a different way doesn't make it an invention.

Yea, guess we both misunderstood. That post was replying to JoeDawg's post suggesting math is an invention. It is in fact, BECAUSE math may be realized in just about any way imaginable, yet the concepts can all be found to be "isomorphic" (if I may use one of your own favorite words) that mathematical concepts must have some basis which is independant of the language and therefore 'discovered'.

 

[Q_Goest]

I imagine, if we had an operational definition of the verbs 'to invent' and 'to discover', there wouldn't be any debate over the answer to the titular question.

If we allow everyone to color those words with their own personal biases, we have no hope of getting anywhere!

Rather than try to define the terms "discover" and "invented" I think a more useful thing to do would be to try and determine how many catagories the natural world might fit into. That's essentially what Penrose would like to do. He breaks the natural world up into the following three:

1. Physical world: Physical, 4 dimensional world. Includes mass, energy.
(Examples include protons, atoms, molecules, energy, cars, planets, people, galaxies, etc... )

2. Mental world: Not objectively measurable, so it doesn’t fit into the physical world.
(Examples include: the redness of an apple, the sweetness of sugar, love, hate, pain, etc...)

3. Platonic Mathematical world: Contains relationships that are 'perfect'.
(Examples include: mathematical operations (such as +, - and =), the mandelbrot set, ratios such as pi and e, etc...)

ok, the last one may or may not be an additional category, and I certainly haven't defined any of these very well, and I don't think my examples of the mathematical world are the best. <sigh>

But the question of "discover" or "invented" can be changed to one of catagorization. Are there more catagories than 3? Should there be less? Or is 3 and only 3 the perfect number?

(See also post #32 for further references to Penrose, Mazure and U of O)

 

[JoeDawg]

Okay then, if that's not math, what is it? What is math describing?

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The same thing english and russian are describing, things in the world. Some of those things appear to be objects, some are relationships between objects. Numbers and functions... Mathematics is simply more abstract and in some ways more precise, because we use it differently for different reasons.

 

[JoeDawg]

Hi JoeDawg,
What do you think of Penrose's suggestion:

... only a small part of the world of mathematics need have relevance to the workings of the physical world.

Only a small part of english need have relevance to the workings of the physical world.
We can talk of unicorns, gods, ETs, and Elves. We can talk about 'fiction' in english, we can even talk about 'nothingness', and some philosophers do, tediously. But these are merely imaginative recombinations of what is 'in the physical world'.

The point is, these are all different languages which describe the same thing, so I'm not sure we can say it is "invented" since others in distant reaches of the universe, could similarly come up with different mathematical languages which would describe the same mathematical concepts we have which, as Penrose notes, have no relevance to the workings of the physical world.

But that's equating the concept of something with the actual thing. A 'triangle' is no less an abstract object than a 'house' is. Both can exist in the real world, but never quite the way we can imagine them. All living creatures would likely need some sort of shelter, so across the universe, aliens probably have 'houses' or something like them.

Our imagination allows us to create many things not in the world. Why would mathematics be different from other languages. And just because 'the house" and "la maison" exist in the real world and describe similar things, doesn't mean that "El Chupacabra" and "the goat sucker" also do. I agree with what Penrose is saying about math, but I don't agree with his conclusion, which I feel is likely based on his love of math. Mathematicians are not the only ones to make this mistake, Philosophers have fallen into linguistic traps too, presuming that concepts in their language are universals in a separate reality. Its an old problem.

 

[Q_Goest]

Hi JoeDawg,
You make a very valid point. Thanks. Unfortunately, I think I've slightly misrepresented what Penrose is saying here. Or at least I've not been faithful to his argument.

Section 1.3 of his book is entiteld "Is Plato's mathematical world 'real'?" Penrose points out that math is robust such that others can verify mathematical truths. Mathematical truths can be agreed to (those parts external to nature that we talked about) regardless of cultural background. Note that this is not particularly true for language as you point out. We could describe various sciences, especially the soft sciences such as psychology or philosophy, in different ways just like blind men touching the proverbial elephant describing different parts. Or we can talk about Unicorns, ET's, elves, etc...

So when Penrose is saying, "... only a small part of the world of mathematics need have relevance to the workings of the physical world." he's not talking about something ficticious, he's talking about mathematical truths (he uses Fermat's Last Theorem as an example) that could be verified by alien life forms billions of light years away. Those same life forms could come up with the same fictious stories about unicorns or elves, but that isn't really the point Penrose is making. He's talking only about mathematical truths which have no relationship, no bearing whatsoever on the physical world.

 

[JoeDawg]

Hmmm. That does make the point clearer. Well, I'm still not sure I agree, but my level of math is rudimentary by comparison. I will say the theorem described strikes me more as a logic problem, or rather one describing the syntactical limits of the language of mathematics, ie given a certain starting point.

In an English sentence, we have an order of: subject(a), verb(b), object(c). Of course people fudge this all the time. By contrast though, mathematicians don't fudge, or consider it wrong if you fudge on agreed upon axioms or syntax.

If we and they were to be as strenuous with natural language however, an alien species would be just as limited as we are, in the ways they could say something... even if the language they spoke wasn't english, but rather a language similarly ordered. The rigid logical structure would rule. It would be inviolate.

Mathematics is much more precise and rigid, than other languages. Apply that to english and you would be accused of being an insane-grammar-nazi. But it empowers a level of logical thinking that natural languages simply can't touch.

In the end it may be my objection to Penrose is more about what he is indirectly implying, rather than what he is trying to say.

 

[CaptainQuasar]

The same thing english and russian are describing, things in the world. Some of those things appear to be objects, some are relationships between objects. Numbers and functions... Mathematics is simply more abstract and in some ways more precise, because we use it differently for different reasons.

Ah, but English and Русскии Язык may also describe things that are not in the world, as I said like Frodo Baggins or slapstick comedy. In those cases both the description and the thing described are invented, ньет? The question is whether mathematics is more like one of those things or if it's something more like gravity.

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[oldman]

But then, shouldn't you be willing to assert that every branch of science is also invented? Because a science never is what it describes...at least things like biology and chemistry and physics...The real question, I think, is whether it's invented in the same way that something like human culture is invented - “of whole cloth,” as it were

I've been going back over this thread, which is now getting a bit involved, and find it's clarified my thinking -- or at least modified my muddled mulling. Thanks for the help, folks.

In it's early stages people like Drachir (#2), Morodin (#4) and HallsofIvy (#5) claimed briefly that Mathematics is both discovered and invented. I now think there's some truth in this. Then Cap'n Q (#14) took the stance that Mathematics is entirely discovered, whereas my view (e.g. bolstered by Joedawg (#10) and Q_Goest's analysis (#32) had been that it's entirely invented. I now think that "entirely" is inappropriate; Cap'Q 's argument about geometrical ratios being discovered is persuasive.

Which brings me to this reply. It is certainly helpful to consider the nature of biology, chemistry and physics as well as mathematics, Cap'n Q. I would say that biology is pretty much entirely discovered, as are most aspects of chemistry. Physics seems to me substantially discovered -- invention (theory) and observation (discovery) have progressed hand in hand for a long while now. But now that its experimental frontiers have become inaccessibly extreme, physics includes large dollops of pure invention --- just look at string theory --- hard to deny that this is invention!

Cosmology, on the other hand, for a long time mostly invention, is finding it also has a robust supporting skeleton made of discoveries like the WMAP results. Just like old fishy Smith discovering the Coelocanth. And what about mathematics? Number theory and algebras seem to me proper examples of invented stuff. But there are aspects of geometry that seem to be discovered, like geometrical ratios (thanks, Cap'n), and statements like " the geometry of space sections is not Euclidean", lately found to be true (the images of distant galaxies are distorted by gravitational lensing). So I now accept that the early posts in this thread were correct -- mathematics is mostly invention, but also somehat discovered.

 

[CaptainQuasar]

And what about mathematics? Number theory and algebras seem to me proper examples of invented stuff. But there are aspects of geometry that seem to be discovered, like geometrical ratios (thanks, Cap'n), and statements like " the geometry of space sections is not Euclidean", lately found to be true (the images of distant galaxies are distorted by gravitational lensing). So I now accept that the early posts in this thread were correct -- mathematics is mostly invention, but also somehat discovered.

Ah! Now one other aspect of what I'm theorizing is that in the same way that set theory and graph theory are isomorphic, geometry and the other branches of mathematics are probably isomorphic to a deeper degree than humans can perceive. So I think that every postulate and deduction within number theory and algebras must also be expressible and provable in geometry as well!

(Though I don't think it's actually even geometry that is “real”, I think the real stuff is probably even more fundamental and something we might not even be able to recognize as akin to geometry.)

So even though number theory and algebra appear to humans to be expressing things that are too abstract to be directly connected to reality, they are in fact proving and specifying things that do identify congruences and structure in the real world.

But now that its experimental frontiers have become inaccessibly extreme, physics includes large dollops of pure invention --- just look at string theory --- hard to deny that this is invention!

I would agree with you that those parts of physics are invented since they must be trying to attribute untrue structures and properties to the real world. But the associated math isn't casting aspersions, it's simply determining self-consistent internal structure which, as I said, through isomorphism must be equivalent to something, somewhere in the real world. So I would say the math is actually more real than the physics!

Ha ha! Take that, physics!

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[oldman]

Ah! Now one other aspect of what I'm theorizing is that in the same way that set theory and graph theory are isomorphic, geometry and the other branches of mathematics are probably isomorphic to a deeper degree than humans can perceive. So I think that every postulate and deduction within number theory and algebras must also be expressible and provable in geometry as well!

You may be right here. Do you know "Geometrical methods of mathematical physics" by Bernard Schutz? He reinforces what you say. BUT -- has it struck you that, roughly speaking, geometry may well be real and to-be-discovered, while dialects like coordinate geometry, the ideas of manifolds etc. are invented to describe this reality, just as there seem to be inventions (languages, algebras) that describe this and other "realities".

Though I don't think it's actually even geometry that is “real”, I think the real stuff is probably even more fundamental and something we might not even be able to recognize as akin to geometry.

Maybe so, but I hope not. There are already enough mysteries to go around: QM for example.

So even though number theory and algebra appear to humans to be expressing things that are too abstract to be directly connected to reality, they are in fact proving and specifying things that do identify congruences and structure in the real world.

Yes, perhaps this is what I'm also suggesting.

I would agree with you that those parts of physics are invented since they must be trying to attribute untrue structures and properties to the real world. But the associated math isn't casting aspersions, it's simply determining self-consistent internal structure which, as I said, through isomorphism must be equivalent to something, somewhere in the real world. So I would say the math is actually more real than the physics!

In this case I agree strongly. There can be nothing wrong with the maths of string theory, but the physics looks very shaky, due to the lack of contact with prediction and verification. We are too given to building towering logical structures on foundations of sand, even when the mortar of logic which makes them cohere seems sound. For example the are a multitude of faiths that can't all be right. Yet they cohere, fiercely.

 

[CaptainQuasar]

You may be right here. Do you know "Geometrical methods of mathematical physics" by Bernard Schutz? He reinforces what you say.

I'm not familiar with it, I'll look it up.

BUT -- has it struck you that, roughly speaking, geometry may well be real and to-be-discovered, while dialects like coordinate geometry, the ideas of manifolds etc. are invented to describe this reality, just as there seem to be inventions (languages, algebras) that describe this and other "realities".

Yes, certainly. I pick geometry because its elements are less like pure symbols, as those of conventional algebra are, so it's more external. But they are still simply constructs to facilitate human thought, still intermediate to the real things they mimic. That's why I think there's probably something more fundamental that comprises the “real” stuff.

I think the concept of Platonic forms must have some truth or meaning to it, at least in the case of mathematics, because something like π is a commonality between many unconnected, disparate things. It seems like the “objects” of mathematics, like a circle or a vector field or a manifold, are really condensations of some diffuse generality, the way we sometimes speak of gravity as an “it” but other times speak in terms of “the law of gravity”. (And I would be saying that the diffuse generalities are what is more real whereas the Platonic condensations are an artifact of human understanding of it.)

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[CaptainQuasar]

I just had an interesting thought… it concerns the way I was saying that the circles which exist in reality as sections of spheres, as orbits, etc. aren't precisely geometric circles, aren't perfect circles, but instead have properties that converge upon those of a geometric circle? If superimposed upon one another the things in the real world which we would analyze with our mathematics would display a distribution of near-circular shapes around the locus, the center-line of a perfect circle.

That harkens to Plato's conception of the physical objects in a particular category being many shadows of a perfect Form that transcends them all. As though behind all of the chickens and mountains in the world there's a perfect über-chicken and a perfect über-mountain existent on a higher plane of reality.

Well, it just occurred to me that this relationship, of real thingie to über-thingie, is something of a parallel relationship to that of the modern QM “particle¹” that doesn't really have position or momentum and is rather a cloud of probability to the classical ballistic cueball-like particle that does have position and momentum.

I don't think this necessarily means anything profound. It's just an interesting connection I thought of.

¹ I put particle in quotes here because I think it was a poor choice for physics to not come up with a new term in the advent of QM. It seems to confuse lots of people particularly since the phrase particle/wave duality is still floating around.

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[oldman]

I just had an interesting thought…

I think this is the forum for such thoughts. I'll mull at yours.

Some time ago I had vaguely similar ideas, and posted https://www.physicsforums.com/showthread.php?t=124737" in the Quantum forum, which got exactly zero responses. Wrong forum, perhaps, or silly ideas. I think that folk who post in the Quantum forum are either baffled newbies to the subject, or busy and polite practitioners of QM (grad students?) too well-schooled in the subject to worry about its foundations.

In the meantime, remember that NOBODY, but nobody, yet truly understands QM.

 

[CaptainQuasar]

Another thing about QM is that in trying to firmly say something about its fundamental meaning it's very easy to make a statement that is easily disputed with evidence.

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[oldman]

... the things in the real world which we would analyze with our mathematics would display a distribution of near-circular shapes around the locus, the center-line of a perfect circle...this relationship... is something of a parallel relationship to that of the modern QM “particle¹” that doesn't really have position or momentum and is rather a cloud of probability to the classical ballistic cueball-like particle that does have position and momentum.


¹ I put particle in quotes here because I think it was a poor choice for physics to not come up with a new term in the advent of QM. It seems to confuse lots of people particularly since the phrase particle/wave duality is still floating around.

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You couldn't have touched on a better example of how a choice of an invented mathematical dialect (wave mechanics) has, in my opinion, resulted in endless confusion (particle/wave duality). in describing a quite simple phenomenon . See the https://www.physicsforums.com/showthread.php?t=124737" I referred to.

 

[Holocene]

Correct me if I'm wrong, but doesn't physics rely heavily on mathematics to accurately describe/analyze some very important principles?

If so, how can we claim to have any respectable handle on physics if our methods for analyzing it are merely "invented"?

 

[oldman]

Correct me if I'm wrong, but doesn't physics rely heavily on mathematics to accurately describe/analyze some very important principles?

If so, how can we claim to have any respectable handle on physics if our methods for analyzing it are merely "invented"?

You are quite correct about physics. It does rely on maths. Physics describes/analyses with language: sometimes it's an ordinary language like English. Often the description of a physics process/phenomenon also uses the language of mathematics.

Now all languages are invented things: think of the one you use and where it came from. Using maths makes the description quantitative and helps predict the future of the process. For instance one might want to explain or predict the colours of light emitted by an atom. To do this you use Quantum Mechanics, which may itself employ different mathematical dialects, like wave equations and wave functions, or operators, matrices and vector spaces. These dialects were also invented. Remember that a language is not the same as the things it describes. People tend to forget this.

 

[CaptainQuasar]

You couldn't have touched on a better example of how a choice of an invented mathematical dialect (wave mechanics) has, in my opinion, resulted in endless confusion (particle/wave duality). in describing a quite simple phenomenon . See the https://www.physicsforums.com/showthread.php?t=124737" I referred to.

Wave mechanics isn't a misnomer though, it really describes the mechanics of waves: ocean waves, sound waves, light waves, radio waves, the mechanical behavior of springs, ensemble phonon behavior in lattices, the slinky-like contractions and expansions in car spacing during traffic flow that civil engineers need to study for building highways and roads - anything that is cyclical or periodic.

The particle/wave duality was a genuine quandary in physics before the advent of QM. The terminological error was that after the advent of QM, once physicists knew that the itty-bitty things they were studying were neither ballistic particles nor waves in some medium, they should have come up with another term. But they didn't - they kept talking about particles, and nicknamed the Schrödinger equation the "wave function" because it's all curvy, even though trigonometric functions don't appear within it anywhere.

I think another thing that perpetuates the problem is, paradoxically enough, that the double slit experiment is so easy to perform. I remember doing it in public school when I was ten or eleven. And of course it's going to be performed at exactly the point you'd be studying the particle/wave duality, so of course that makes the subject stick more firmly in childrens' minds.

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[a2tha3]

Well in order to fully try to answer this question to the best of my ability, I think you should ask yourself and others whether anything is discovered or invented, and most likely you will get contradictory opinions. It's all a matter of perspective and in a religious perspective everything is invented (Just to be clear I'm agnostic) by 'God' and from a scientific perspective everything is discovered. Religion ultimately suggests that god is the source of everything, so that explains why people would pick invent, and science is all about discovery. The people who pick both (religious scientifics?) do so to attempt to extend their openess iin thier

 

[CaptainQuasar]

Well in order to fully try to answer this question to the best of my ability, I think you should ask yourself and others whether anything is discovered or invented, and most likely you will get contradictory opinions. It's all a matter of perspective and in a religious perspective everything is invented (Just to be clear I'm agnostic) by 'God' and from a scientific perspective everything is discovered. Religion ultimately suggests that god is the source of everything, so that explains why people would pick invent, and science is all about discovery. The people who pick both (religious scientifics?) do so to attempt to extend their openess iin thier

I'm an atheist myself but I will say that this is a completely false dichotomy. Religious and scientific are not opposites. Atheists who think that all of their thoughts derive from rationality are fooling themselves. I've met religious people who are far more rational than many other atheists I know.

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