Are you a platonist when it comes to mathematics - do you believe that purely abstract objects exist for mathematical ideas that are both independent?

[Unity+]

I just wanted to take a little poll of everyone's view of the world.

 

Are you a Platonist of mathematics, or someone that thinks that Mathematical structures exist in nature and are abstract, but real forms or do you believe that Mathematics simply is a human construct that allows us to interpret the nature of the world?

Edited March 7, 2014 by Unity+

[studiot]

Since there is evidence that other creatures have rudimentary counting and measurement skills, maths cannot be a purely manmade construct.

Edited March 7, 2014 by studiot

[StringJunky]

Are we extant from Nature?

[John]

Maybe I'm confused. I thought platonism was a form of realism.

[ydoaPs]

Maybe I'm confused. I thought platonism was a form of realism.

It is.

[imatfaal]

per John and yDoaps above - platonism (and you really should spell it with a lower case p as it is not the exact work of Plato) concerns the belief that truly abstract objects exist that are situated in neither space nor time additionally they are also not purely imaginative/mental objects existing only in the mind's eye of the philosopher. They are purely abstract, unchanging, and completely divorced from our reality.

 

The number 14 has (according to p-ism) has an independent existence unchanged, uninteracting, in this abstract realm - things that use 14 in our lumpen reality are instantiations of this ideal abstract form. All things that can be characterised through use of the number 14 have in common that they all, in part, exemplify the abstract ideal.

 

This is such a contingent and complex theory that a polar opposite is hard to define (and it definitely is not just realism) - absolute conceptualism would say that these ideas are only mental/cognitive, immanent realism would argue that all objects are interactable with, changeable and changing, ie not purely abstract. And as there is a reality for these abstract objects to be distinct from you cannot really claim that the diametric opposite of platonism is realism.

 

 

I think the original question can stand if the stated dichotomy is removed -

 

"Are you a platonist when it comes to mathematics - do you believe that purely abstract objects exist for mathematical ideas that are both independent of spatio-temporal reality and of our own mental imaging?"

[Unity+]

per John and yDoaps above - platonism (and you really should spell it with a lower case p as it is not the exact work of Plato) concerns the belief that truly abstract objects exist that are situated in neither space nor time additionally they are also not purely imaginative/mental objects existing only in the mind's eye of the philosopher. They are purely abstract, unchanging, and completely divorced from our reality.

 

The number 14 has (according to p-ism) has an independent existence unchanged, uninteracting, in this abstract realm - things that use 14 in our lumpen reality are instantiations of this ideal abstract form. All things that can be characterised through use of the number 14 have in common that they all, in part, exemplify the abstract ideal.

 

This is such a contingent and complex theory that a polar opposite is hard to define (and it definitely is not just realism) - absolute conceptualism would say that these ideas are only mental/cognitive, immanent realism would argue that all objects are interactable with, changeable and changing, ie not purely abstract. And as there is a reality for these abstract objects to be distinct from you cannot really claim that the diametric opposite of platonism is realism.

 

 

I think the original question can stand if the stated dichotomy is removed -

 

"Are you a platonist when it comes to mathematics - do you believe that purely abstract objects exist for mathematical ideas that are both independent of spatio-temporal reality and of our own mental imaging?"

I meant all of this. When I was watching a film about the two philosophies, they referred to the idea of numbers existing in reality as "platonism" while they referred to realism as being numbers only existing in the mind of humans.

 

EDIT: The poll has been reworded. I apologize for the confusion.

Edited March 7, 2014 by Unity+

[John]

In Letters to a Young Mathematician, Ian Stewart claims "...the working philosophy of most mathematicians is a mostly unexamined Platonist-Formalist hybrid."

 

While I'm still a student and I would hesitate to really call myself a "mathematician" yet, I think this is where I fall as well, perhaps with a bias towards the platonist side of that coin. As my education continues, depending on how far down the rabbit hole my ultimate area of focus takes me, perhaps I'll become more concerned with the philosophical foundations of my work. Of course, I may just end up with a thoroughly examined platonist-formalist hybrid at the end.

Besides not really being a mathematician, I also spend my time not being a philosopher of mathematics, but with my current understanding, I don't find the schools besides platonism and formalism very appealing or convincing (and even formalism loses some luster in light of the incompleteness theorems).

All this is to say: I guess my answer is "Yes."

Edited March 8, 2014 by John

[Unity+]

In Letters to a Young Mathematician, Ian Stewart claims "...the working philosophy of most mathematicians is a mostly unexamined Platonist-Formalist hybrid."

 

While I'm still a student and I would hesitate to really call myself a "mathematician" yet, I think this is where I fall as well, perhaps with a bias towards the platonist side of that coin. As my education continues, depending on how far down the rabbit hole my ultimate area of focus takes me, perhaps I'll become more concerned with the philosophical foundations of my work. Of course, I may just end up with a thoroughly examined platonist-formalist hybrid at the end.

 

Besides not really being a mathematician, I also spend my time not being a philosopher of mathematics, but with my current understanding, I don't find the schools besides platonism and formalism very appealing or convincing (and even formalism loses some luster in light of the incompleteness theorems).

 

All this is to say: I guess my answer is "Yes."

I think most mathematicians would identify themselves, like me, as a platonist. It seems, without a doubt, that nature has constructs of mathematics based on mathematics. Of course, this could be caused by the limitation of the human mind.

[ajb]

I have no idea which is "true" nor do I really think it is a question that can be properly answered.

 

However, my opinion based on observations of mathematics and mathematicians is that I follow some form of platonism. This means I discover mathematics rather than invent mathematics, but again the distinction is not very clear.

 

I justify this as follow...

 

One school of thought is that mathematics is just a game in which one can make up the rules from the start in anyway you see fit. However, the truth is that mathematicians all round the world are not working on abstract and independent systems, but rather mathematics has evolved into interlaced branches. Not all abstract systems have "nice properties" and not all are "interesting". Thus it seems to me that there is some larger set structure here that we are probing. You cannot just make up the rules and end up with interesting non-trivial structures.

 

But this is just my pragmatic philosophy and I am not sure how much it really drives my research, but for sure I would not like to study very abstract systems for their own sake and everything should be well motivated from already studied systems.

[imatfaal]

Hmm - to investigate the two above ideas by Unity and John - I would present a little scenario. Platonism requires that these abstract objects are unchanging in space and time - they remain inviolate (this is what removes it from the work of the man Plato who talks of interaction and influence).

 

A few centuries ago there was a clear conception of parallel lines founded in Euclid around which much of geometry, and thus greek mathematical and thus most of mathematics was based; which if platonism is correct should remain unchanged. Well now most mathematicians have a new concept of parallel lines after Gauss and Bolyai - so either the unchanging inviolate sacrosanct (etc.) ideal has been lost and replaced or it has be appended with a tag saying "this is not quite right". I would argue this means that objects which are held to be abstract ideals can be altered. So the space for platonism to remain valid is for those abstract ideal objects which are completely correct in all terms, all axiomatic bases and all times - and that is practically nothing.

 

As mathematics is contingent upon man-made axiomata, and that no sufficiently complex system of mathematic logic can be entirely self-consistent where is the philosophical phase space within which platonism can exist. Absolutely abstract ideals cannot be contingent upon the starting points of each man-made argument - yet what in modern mathematics is not founded on a more basic assumption. And if ideals exist then surely we can construct ideals through the work of Godel that are false propositions yet correctly formed and internally contradictive. An ideal that viewed from one perspective is acceptable and from another is fallacious seems to be intimately tied to its creators in a way that abstract ideals cannot be.

[studiot]

 

I have no idea which is "true" nor do I really think it is a question that can be properly answered.

 

 

 

There is no requirement for either to be 'true', other possibilities exist.

[cladking]

Everything is a construct except reality itself. Even reality can become a construct to an individual if he defines everything in terms of other constructs. We are what we believe and to a very real extent the world is what an individual believes. We understand and format reality and our knowledge in language which can be a kind of construct as well. Symbolic language like we use seems especially prone to the creation of such constructs as we understand things symbolically and as models of reality.

 

We believe there are no alternatives.

 

The reality is that numbers have no external existence to the individual. Math is a highly formal logic and highly useful tool to calculating and understanding nature but has no existence. If you see six elephants walk by they will each be different and each will be unique. One might be old and walking its final steps and one might be ready to birth. If the aged one dies on its way by then how many walked by? If one is born then how many went past? Any such event will require some time so even the concept of "walking by" is shown to be a sort of construct as well. There were six elephants and if one simultaneously dies as another is born then there are six different elephants though at least four and perhaps five are virtually identical.

 

What if ten lemmings go by but you see only six? How about a school of 500 fish that look like 5000? The three miles to the nearby town is never the same in both directions or from day to day. Nothing in reality is repeatable or countable.

[Unity+]

Hmm - to investigate the two above ideas by Unity and John - I would present a little scenario. Platonism requires that these abstract objects are unchanging in space and time - they remain inviolate (this is what removes it from the work of the man Plato who talks of interaction and influence).

 

A few centuries ago there was a clear conception of parallel lines founded in Euclid around which much of geometry, and thus greek mathematical and thus most of mathematics was based; which if platonism is correct should remain unchanged. Well now most mathematicians have a new concept of parallel lines after Gauss and Bolyai - so either the unchanging inviolate sacrosanct (etc.) ideal has been lost and replaced or it has be appended with a tag saying "this is not quite right". I would argue this means that objects which are held to be abstract ideals can be altered. So the space for platonism to remain valid is for those abstract ideal objects which are completely correct in all terms, all axiomatic bases and all times - and that is practically nothing.

 

As mathematics is contingent upon man-made axiomata, and that no sufficiently complex system of mathematic logic can be entirely self-consistent where is the philosophical phase space within which platonism can exist. Absolutely abstract ideals cannot be contingent upon the starting points of each man-made argument - yet what in modern mathematics is not founded on a more basic assumption. And if ideals exist then surely we can construct ideals through the work of Godel that are false propositions yet correctly formed and internally contradictive. An ideal that viewed from one perspective is acceptable and from another is fallacious seems to be intimately tied to its creators in a way that abstract ideals cannot be.

Recent work I have done has relations with this kind of problem, as I think there seems to be a difference between axioms founded upon nature and axioms founded upon the un-natural(for lack of a better word).

 

At the beginning of Mathematics, many mathematicians developed axioms of logic that were related to the reality of nature, not upon the construction of the human mind. Thus, mathematics came to be founded upon the axioms of nature. However, soon enough we begin developing axioms that relate to other systems. Therefore, humans have the capability of discovering the axioms of nature, but they also have the capability of developing axioms of other types of systems.

 

For example, there are many abstract constructs which are used very much in physics, such as many geometries that are abstract and numbers(complex numbers) which are used in quantum mechanics. There are also abstract constructs that do not fit within our CURRENT Universal landscape.

 

In regards to parallel lines, I compare the change to the change between Newtonian mechanics and Einstein's theories of relativity and the forces that exist, such as gravity. Einstein didn't change the particular ideas surrounding the existence of the force of gravity, but completed Newton's ideas as to add a mechanism for the gravitational forces that exist. Therefore, Newton was not wrong but incomplete with his theories.

 

Ideas develop over time. This does not mean that the original idea is wrong, but needs modifications to fit more and more closely to the natural world. Discovery is not one particular event, but a group of events that leads us to understand more fully the mechanisms of the Universe.

[arc]

However, my opinion based on observations of mathematics and mathematicians is that I follow some form of platonism. This means I discover mathematics rather than invent mathematics, but again the distinction is not very clear.

 

 

Are we extant from Nature?

 

 

 

There is no requirement for either to be 'true', other possibilities exist.

 

As mathematics is contingent upon man-made axiomata, and that no sufficiently complex system of mathematic logic can be entirely self-consistent where is the philosophical phase space within which platonism can exist. Absolutely abstract ideals cannot be contingent upon the starting points of each man-made argument - yet what in modern mathematics is not founded on a more basic assumption.

 

Back when I was young and impressionable, Carl Sagan in his Cosmos series said something that forever set in place my way of thinking on these matters. His comment was in regards to and something along the lines of; if for a few key discoveries at the right time in their history, the ancient Greeks could have put a man on the moon first. This observation by Mr Sagan put forth the idea to me that, as ajb expressed, "I discover mathematics rather than invent mathematics"

 

This seems to be the correct view of how we have acquired our knowledge of mathematics. We find things, we stumble upon them or systematically search them out, but if we do not, someone else will in due time. These ideals or structures seem to date to the beginning of time. And I would guess are infinite and contain both abstract (to us) and real (to us) mathematical structures.

 

We currently issue thousands of letters of patent every year to individuals who more realistically discovered their ideas rather than invented them out of nothing as ones ego might insist. And to strengthen this point, they no doubt worried that someone else might beat them to it first. And too, by using Mr Sagan's perspective, it could even have been the Greeks.

 

We see in nature that working concepts, based in or simply defined by mathematics, are at the foundation of evolution. Nature has searched out the optimal wing for the specific time and application in numerous occasions. We have discovered these concepts, these principles of flight ourselves and refined them through time with our own systematic search of the mathematical realm. We continually discover a more "optimal" airfoil design that no doubt has a certain elegance in its maths, it rings true so to speak with nature and mathematics both, just as it has in the wings of a bird.

 

One might even say with a certain harmony between the observable world and the unseen mathematical fabric of the universe.

Edited March 9, 2014 by arc

[imatfaal]

Recent work I have done has relations with this kind of problem, as I think there seems to be a difference between axioms founded upon nature and axioms founded upon the un-natural(for lack of a better word).

 

At the beginning of Mathematics, many mathematicians developed axioms of logic that were related to the reality of nature, not upon the construction of the human mind. Thus, mathematics came to be founded upon the axioms of nature. However, soon enough we begin developing axioms that relate to other systems. Therefore, humans have the capability of discovering the axioms of nature, but they also have the capability of developing axioms of other types of systems.

 

For example, there are many abstract constructs which are used very much in physics, such as many geometries that are abstract and numbers(complex numbers) which are used in quantum mechanics. There are also abstract constructs that do not fit within our CURRENT Universal landscape.

 

In regards to parallel lines, I compare the change to the change between Newtonian mechanics and Einstein's theories of relativity and the forces that exist, such as gravity. Einstein didn't change the particular ideas surrounding the existence of the force of gravity, but completed Newton's ideas as to add a mechanism for the gravitational forces that exist. Therefore, Newton was not wrong but incomplete with his theories.

 

Ideas develop over time. This does not mean that the original idea is wrong, but needs modifications to fit more and more closely to the natural world. Discovery is not one particular event, but a group of events that leads us to understand more fully the mechanisms of the Universe.

My problem with your argument is that you are simply talking about mathematical ideas being discovered rather than the platonism idea that these abstract objects are inviolate and unchanging. Platonism (as opposed to the ideas of Plato the man) requires that these objects are temporally and spatially unvarying. So with either the example of parallel lines or that of gravity we have to (if platonism is correct) posit that there exist various abstract objects that are wrong or incomplete (ie Newtonian gravity or Euclidean parallel lines) OR that the only abstract models that exist are perfect (I don't think any scientist really believes this). You cannot have developing abstract ideals in platonism.

[Sato]

Is mathematics not just an emergent property of biology and interacting neurons like any other form of consciousness, perception, and understanding?

[ajb]

This seems to be the correct view of how we have acquired our knowledge of mathematics. We find things, we stumble upon them or systematically search them out, but if we do not, someone else will in due time.

This I think is absolutely true of mathematical research, which is what I base most of my philosophy on. However, that is not the same as mathematics itself.

Edited March 13, 2014 by ajb

[arc]

This I think is absolutely true of mathematical research, which is what I base most of my philosophy on. However, that is not the same as mathematics itself.

 

ajb, Could you give me a little more detail of that difference between the mathematical research and the mathematics itself. I'm rather a lightweight in philosophy so I don't quite understand the separation on the discovery vs. ? aspect, thanks.

[ajb]

ajb, Could you give me a little more detail of that difference between the mathematical research and the mathematics itself.

What I mean is that resarching mathematics often involves lots of "experiments" via the construction of examples + simple cases, some guess work and following hunches. One seems to probe these structures before finding the proper generality one is looking for as well as working on top of well established constructions, usually. Now this is often very different to how mathematics is presented, which is usually theorem and then examples.

 

In this sense I think mathematics is discovered, but really I am talking about how mathematicans work rather than the mathematics itself.

[hoola]

certainly ! I see reality as being derived from information that was cobbled together by the conjunction of the platonic spherical structure of the singularity (it's circular cross-section having the circumference/diameter of PI), and the mathematics of that endless string of numbers eventually describing this universe, starting with the big bang...although I don't believe that they are "independent"...the spherical singularity came into existence and remained complete, but the PI is ongoing and never complete...yet they started at the same moment, as determined by logic for their continuing existence...

Edited April 21, 2014 by hoola

[michel123456]

To me, mathematics are emergent from geometry.

I don't know where geometry comes from. It should also be an emergent feature. Not from numbers anyway. Numbers can be arranged geometrically (in a line for example), not the other way round.

[hoola]

I see that the first digit of math proceeded from the void, as there was only ONE void...there is your seed of information. That one digit has a theoretical physical collary of the point, or sphere, which has the circumference/diameter ratio that expresses the endless string of 3.14159.... regardless of size, even if that size is the dimensionless point......that constructs all maths. That math construction goes on for trillions of years forming an Informational Black Hole ( IBH ) which projects as a hologram, our universe....

[Unity+]

I see that the first digit of math proceeded from the void, as there was only ONE void...there is your seed of information. That one digit has a theoretical physical collary of the point, or sphere, which has the circumference/diameter ratio that expresses the endless string of 3.14159.... regardless of size, even if that size is the dimensionless point......that constructs all maths. That math construction goes on for trillions of years forming an Informational Black Hole ( IBH ) which projects as a hologram, our universe....

I don't know if you were being serious...

 

To me, mathematics are emergent from geometry.

I don't know where geometry comes from. It should also be an emergent feature. Not from numbers anyway. Numbers can be arranged geometrically (in a line for example), not the other way round.

Theoretically, any form of mathematics can be represented as a particular set of geometry, whether it be the known form of geometries or unknown. It is an interesting question dealing with whether the numbers are separate from the geometry or if they are conjoined together.

[ajb]

Theoretically, any form of mathematics can be represented as a particular set of geometry, whether it be the known form of geometries or unknown.

Can you explain that a bit further?

 

I remember asking Sir Michael Atiyah a similar question; "what is geometry?"

 

His reply was more or less that geometry is algebra when you can think in terms of pictures. This I liked as modern geometry like algebra in which you can think geometrically, often with some abuse of language.