On math being between sensory-based and more abstract levels of human thought

[Kyriakos]

For lack of a better thread title... this is a brief synopsis of the views on what Math is, in the levels of less to more abstract thinking, as presented mostly by Plato in a couple of his dialogues (eg Parmenides and The Republic).
The topic for discussion in the thread is to theorise on where math is to be placed, in regards to a scale of more abstract to less abstract human thinking subjects. You can type your own view, and i hope there can be some discussion...
The reason for this little presentation is that (as usual) it is part of my seminars, so i thought it would be good to practise it here a bit, given that if i can (decently??) present it in English, it follows it will be far more flowing in my superior native language

First some background on math of that era:

The two first math theorems were that of Thales (about a right-angled triangle inscribed in a semi-circle), and Pythagoras (the famous hypothenuse of a right-angled triangle one). Plato's time was around 2 centuries after Pythagoras, and almost 3 after Thales. In his Academy (a school for philosophy) Plato focused mostly on geometry as a means of examining how correctly and intricately a student could think. But in his dialogues (eg in the Cave Allegory in the 7nth book of the Republic) he names Math as only the lowest level of the higher (less sensory-bound, more abstract) calculations one can work with, and places it below all pure Ideas/Archetypes, and of course also below the edge of his system, which is some over-Idea that shines over them like a Sun, allowing humans to examine them all..
At the time of Socrates math had already moved massively to calculations of irrational numbers (the so-called Spiral of Theodoros is one of the main subjects of the Socratic Dialogue titled Theaetetos or On Science), primes and symmetries, and of course just 80 years after Plato we have the likes of Eukleid and Archimedes and many other towering mathematicians.

An overview of the argument about Math being less abstract than pure Ideas

1)Math has the unique element in all human studies that it features forms which are the same for any human observer. Eg if i tell you i see a (perfect) circle, and you know what a circle is, there is no way that you will imagine some other form apart from the circle. On the other hand if i told you that i am watching a tree, a chair, a human etc, you could never imagine the exact same, cause those are not set in oneness of form, but are multitudes which all have the same overall name/category.
2)Math, also, is something which developed from axioms. An axiom literaly means something which seems 'self-evident' and thus can aspire to be held as true without any proof given. Eg if one says that a human is One human, and not Two, that can be taken as axiomatically and evidently true. If one claimed they have 5 pens in their pocket... that would not have to be true and we would need proof.
Given humans can sense evidently some sums as integers, the progression of integer numbers is in many ways the basis of math, not just because the first positive number (1) is also a Meter of all numbers, but moreover due to properties studies in this progression (for example primes, fibonacci numbers, etc). The basic progression is 1, 1+1, 1+1+1,... This already sets the tone for math in regards to crucial notions such as the primes, cause they too are formed by examining if there are perfect divisors of a number.

3) Math has set basic forms and notions, which rest on axioms, and thus is very different from purely abstract ideas such as terms which signify not a specific and unigue image/being/subject, but anything which is partly or crucially defined by them as well. A leg is a term which signifies a moving limb used to sustain something while it touches the surface of movement. But the term itself does not provide us with set info on a particular instance of this. It can be anything from a human leg to the scales of some arthopod. Which brings us to the main argument against math being the highest abstraction:

4) Plato/Socrates noted that we cannot actually define "Knowledge" (The Theaetetos dialogue), cause for that to be possible we would need to set our definition on a final, atomic, non-breaking into more subcategories, stable substrate of the object we claim to have knowledge of (either a mental or material object, btw). Contrary to that we can have math terms which remain eternally the same, and can be examined without much difficulty by a student, cause there we already set the end of that substrate, through our axioms.
Plato ultimately is of the view that math is like a ladder which can help up to some point, but ultimately itself is not of the same type as the highest orders of thought. Chaos is a ladder too, but anyway, i think i will end this OP here since i suppose already almost no one will try to read it...

[Nikitn]

How abstract math is, depends entirely on the math. Some math deals with real-life problems,, other very very formal and general maths can deal with entirely abstract concepts that have no relation to the real world what-so-ever.

Also the view of the ancient philosophers is pretty irrelevant on this, as they only knew elementary geometry and arithmetic, which are very concrete forms of mathematics.

[Kyriakos]

How abstract math is, depends entirely on the math. Some math deals with real-life problems,, other very very formal and general maths can deal with entirely abstract concepts that have no relation to the real world what-so-ever.

Also the view of the ancient philosophers is pretty irrelevant on this, as they only knew elementary geometry and arithmetic, which are very concrete forms of mathematics.

But all math follows from set axioms, so why do you claim the latter math is inherently different in this manner? (ie more abstract/less axiom-based, existing in a less axiom-based field of examination etc).

Btw by the time of Plato math was not as elementary as you (i think) seem to claim it was. Eg Pi was already studied in square root hypothenuses of a spiral (the following one: )



Let alone that only 80 or so years after Plato we have the publication of The Elements, by Eukleid, and all kinds of intricate treatises in the Greek world (proto-calculus by Archimedes using a hybrid of trigonometry with his 'Lever' mechanics, or Apollonios of Perga and Erathosthenes with works on prime numbers and 3d forms such as cones, along with the calculus examination by Archimedes of the identity between the volume of a Cylinder and the same-diameter volumes of a Sphere added to a Cone). So i do not see why you suggest the math of that era was so primitive, nor that it had nothing to do with the epicenter of the philosophy of that time, given that math was the main core of presocratic philosophy (infinite series by Parmenides, Zeno and Heraklitos, Protagoras and the idea of the atom and geometric questions about the axiomatically set notion of 'a single point' and so on. All those still are the axioms basing math).

[Town Watch]

The problem seems to exist in two or more different "logics" with which math is to be understood.


Didnt bertrand russel attempt to reduce math to symbolic logic in principia mathematica? Logicism it was called.

It seems that gödels incompleteness theorem runs against such a venture?

The problem seems to be, different logic systems...

Ancient greeks started with classical logic... then came plato and platonic logic (? ) etc... until today in math there exist different logics.

This is my limited knowledge of the problem at hand.

[Himster]

1)Math has the unique element in all human studies that it features forms which are the same for any human observer. Eg if i tell you i see a (perfect) circle, and you know what a circle is, there is no way that you will imagine some other form apart from the circle. On the other hand if i told you that i am watching a tree, a chair, a human etc, you could never imagine the exact same, cause those are not set in oneness of form, but are multitudes which all have the same overall name/category.

While I can't say this is untrue, I can't say I completely agree.
Concerning imagining a specific form like a (perfect) circle, we have learned (in school/from culture) what a circle is, it is a symbol. The same can be applied to other forms, like Tolkien fans could say "imagine the symbol of the white tree of Gondor," those who know what that is would imagine identical forms, the same as any constructed symbol, disseminated in the same way: school/culture/learning etc.

2)Math, also, is something which developed from axioms. An axiom literaly means something which seems 'self-evident' and thus can aspire to be held as true without any proof given. Eg if one says that a human is One human, and not Two, that can be taken as axiomatically and evidently true. If one claimed they have 5 pens in their pocket... that would not have to be true and we would need proof.

I wouldn't be so quick to conclude that mathematics is fundamentally axiom based, I think it is perfectly possible that mathematics could be empirically based, from which we can derive hypotheticals: as you say in your third point (or at least what I think you say in your third point). Now whether there is truly a valid distinction between hypotheticals and axioms in this instance is up for debate.

To clarify what I mean by hypoethicals and empirical in this context: a man claims to have four pens in his pocket, that can be empirical verified. But a man with two pens and then says IF I had two more I would have four pens: that is hypoethical, which is something I don't necessarily equate with axiomatic, but I don't quite see the need to call it hypothetical.

Given humans can sense evidently some sums as integers, the progression of integer numbers is in many ways the basis of math, not just because the first positive number (1) is also a Meter of all numbers, but moreover due to properties studies in this progression (for example primes, fibonacci numbers, etc). The basic progression is 1, 1+1, 1+1+1,... This already sets the tone for math in regards to crucial notions such as the primes, cause they too are formed by examining if there are perfect divisors of a number.

Excellent. Very well put.

3) Math has set basic forms and notions, which rest on axioms, and thus is very different from purely abstract ideas such as terms which signify not a specific and unigue image/being/subject, but anything which is partly or crucially defined by them as well. A leg is a term which signifies a moving limb used to sustain something while it touches the surface of movement. But the term itself does not provide us with set info on a particular instance of this. It can be anything from a human leg to the scales of some arthopod. Which brings us to the main argument against math being the highest abstraction:

4) Plato/Socrates noted that we cannot actually define "Knowledge" (The Theaetetos dialogue), cause for that to be possible we would need to set our definition on a final, atomic, non-breaking into more subcategories, stable substrate of the object we claim to have knowledge of (either a mental or material object, btw). Contrary to that we can have math terms which remain eternally the same, and can be examined without much difficulty by a student, cause there we already set the end of that substrate, through our axioms.
Plato ultimately is of the view that math is like a ladder which can help up to some point, but ultimately itself is not of the same type as the highest orders of thought. Chaos is a ladder too, but anyway, i think i will end this OP here since i suppose already almost no one will try to read it...

Yeah, I don't think I totally understand that, but it sounds good and It's pleasantly much shorter than your usual OPs. So well done there.
Anyway, I shall try to respond, kinda.
The sequence of form to axiom/hypothetical. I would think for very practical reasons physical-form has to have to pre-existed "axiom". The idea that 2+2=4 has to have been derived empirically before it was set as a truism: two bails of hay from the valley added to two bails of hay from the town is four bails of hay, next week IF I get another bail of hay from my cousin, I'll have five bails of hay........ and so on.

[Sphere]

1+2+3+4+5+6.... = -1/12

[Kyriakos]

Thank you for your posts (but not Sphere ).

Given i just got back from the fourth presentation, i will likely reply tomorrow. At last my seminar work this week is over

[Sphere]

Thank you for your posts (but not Sphere ).

It's more pertinent than you might think.

That the sum of all integers can be represented by -1/12 is astonishing. That this result is applicable to physics, and thus the real world, is downright mind bending.

[Kyriakos]

It's more pertinent than you might think.

That the sum of all integers can be represented by -1/12 is astonishing. That this result is applicable to physics, and thus the real world, is downright mind bending.

I recall a video on that, but IIRC --saw it a couple of months ago-- it was based on rather loose logic (i mean can you really add stuff to the end of an 'infinite series'? Even if you break that series up? Doesn't it go against the notion of an infinite series if you add to it? Or at least if you ad integer to infinite set and then detract from another infinite set, etc?). But maybe i saw the crap version of the video/idea (not that unlikely)

[Nikitn]

But all math follows from set axioms, so why do you claim the latter math is inherently different in this manner? (ie more abstract/less axiom-based, existing in a less axiom-based field of examination etc).

Because it has no relation to what goes on in the real world. As for your axioms argument: Yes, maths is based on axioms, but so is our perception of reality itself. After all the definition of the adding/subtracting operation is just as much based on assumptions as us thinking what we see and feel is real.

Btw by the time of Plato math was not as elementary as you (i think) seem to claim it was. Eg Pi was already studied in square root hypothenuses of a spiral (the following one: )



Let alone that only 80 or so years after Plato we have the publication of The Elements, by Eukleid, and all kinds of intricate treatises in the Greek world (proto-calculus by Archimedes using a hybrid of trigonometry with his 'Lever' mechanics, or Apollonios of Perga and Erathosthenes with works on prime numbers and 3d forms such as cones, along with the calculus examination by Archimedes of the identity between the volume of a Cylinder and the same-diameter volumes of a Sphere added to a Cone). So i do not see why you suggest the math of that era was so primitive, nor that it had nothing to do with the epicenter of the philosophy of that time, given that math was the main core of presocratic philosophy (infinite series by Parmenides, Zeno and Heraklitos, Protagoras and the idea of the atom and geometric questions about the axiomatically set notion of 'a single point' and so on. All those still are the axioms basing math).

Compared to today it is very primitive. We have managed to create models of the atoms, theories which describe the shape of the universe and so on. That's a long road away from merely playing with fractions and making geometrical proofs, as impressive as that might be.

1+2+3+4+5+6.... = -1/12

Well as far as I can read on wikipedia, you get that result only if you use very special methods that assign finite values to divergent series. I'm sure if you dig far enough, said methods will start making sense as they're probably just based on defining stuff in new ways.

At any rate, from Calculus we both know that series diverges to infinity.

[Sphere]

At any rate, from Calculus we both know that series diverges to infinity.

By the theory of limits, yes, it is divergent. But why should that be definitive? Is infinity or -1/12 more "correct"? I don't think there is an obvious answer.

I recall a video on that, but IIRC --saw it a couple of months ago-- it was based on rather loose logic (i mean can you really add stuff to the end of an 'infinite series'? Even if you break that series up? Doesn't it go against the notion of an infinite series if you add to it? Or at least if you ad integer to infinite set and then detract from another infinite set, etc?). But maybe i saw the crap version of the video/idea (not that unlikely)

I don't think it is loose logic. I think it just brings into question what we mean by an equals sign.

I brought it up as an example (there are others like it) because it is both highly abstract (we cannot really wrap our minds around the idea that -1/12 can be characteristic of the sum of all integers), but none the less is useful in describing the physical world. Particularly in quantum mechanics where you need to get real answers out of otherwise infinite results (i.e. renormalization). When Nobel Prizes are involved, it becomes difficult to dismiss.

[Nikitn]

By the theory of limits, yes, it is divergent. But why should that be definitive? Is infinity or -1/12 more "correct"? I don't think there is an obvious answer.

I'm not saying it's definitive, I'm saying it's not that big of a deal as it's just based on defining stuff to be this and that. And certainly it's a much more reasonable for the series to diverge than converge to -1/12, from a real-world perspective. Don't get me wrong though, I'm not saying that we we see and feel is exactly how it is,, nature is much more intricate than that.

[Aulus]

I would argue that math can be as abstract as one wishes it to be. Geometry is hardly the most abstract portion of math, as one can work with it through numbers, pictures, models, etc - and yet can be made more abstract by doing geometry in different spaces. The basics of calculus too I would say are fairly concrete as well (especially when used for physics), but can quickly spiral into abstraction. Moving onto topics such as complex analysis and infinite sums and expansions, and it quickly becomes a lot less simple. This is just the math I know requisite for physics, to describe the real world. Go further into the realm of pure mathematics, and you can go down the rabbit hole as long as you want.

I see math as the language science uses to write the Universe. It has its own prose and it has its own poems.

[Kyriakos]

^I also think of it in a similar way, but to me it seems (due to the set axioms being so crucial) that using math is like being set to write a novel in the manner a previous author did, using some key characters (math symbols etc) and other things he also defined, so it is a bit more closed-up than art

Anyway, sorry for not replying at length but i am still detoxing from the seminars this week

But i did post some stuff on the same topic i maintained in another forum too, and given what i typed there is general info on Platonic definitions of the Ideas in some dialogues, maybe i can be excused for just copy-pasting that lazily here too...:

-An "Idea" is not having to be an actual depiction of something. You seem to argue that the idea of a circle is in tautology with a visual imagining of a perfect circle. While some may try to have such a thing (itself quite problematic; are you really imagining something that factors an irrational number so as to form as an image in your imagination?), the idea of a math form following from axioms and a stable definition (as in the case of circle, square, cube, line, point, etc) can be taken as being there by virtue of the set method you have in mind, cause you can communicate it fully and already have the atomic parts of it defined as well (in juxtaposition to what i noted at in my reply to the first part of your comment in this post * ).
-An Idea is in essence a category or type. In fact the original Platonic term is not "Idea", but "Eidos", which means "type" or "kind"/category. The ideas are any separate category/type you can think of. That pretty much includes any notion we have for anything, including all notions we have for examining notions we have, and so on, to endless iterations

The argument by Plato/Socrates on Ideas boils down to the view (which i find correct, at least for the math i am familiar with and they talk about, but i already asked if any more modern math can be said to not flow from axioms as those older math did-- which i doubt is true) that math notions/ideas are formed in clearer and closer dependence on human sensory input, eg geometric notions seem to follow from humans having the ability to move in space, to observe things as integers, etc, while at the same time core math elements are abstractions that create antithesis in the overall system (eg Democritos, the one who theorised on 'atomic' final parts, had noted that the notion of 'single point' creates issues in geometry, since it functions as a sort of atom there but in a system which is not part of nature anyway. There is one paradox left by him, or rather a question, about whether a cone sectioned by a plane will have in two immediate outer points in that section an just above or below it, the same lenght or less/more, and it would follow that if it has less the cone is in reality step-by-step created in its atomic parts, while if it has the same length it would follow that it would remain the same even if we multiply that infinitesimal part, thus we would now have a cylinder instead of a cone).

Basically some notions in math are more tied to sensory or somatic input, others are more on the purely mental side of things (like the notion of Infinity, which the Eleatic philosophers (Parmenides, Zeno, etc) were so fond of using to argue that the senses and human perception in general is only a source of illusion and errror).

*(first part alluded to: ) There is a difference between inherent (ie perpetual by definition) abstraction, and complication, since the latter can (potentially) end at some point (eg if you can reach an atomic part of that which you aim to present/describe/define). In the case of 'abstract ideas' the argument is that there is no atomic end to an attempt to define a specific manifestation of them, so the idea itself is entirely abstract. The dialogue "Parmenides, or on Ideas" focused on the apparent inability to "know" something in the realm of the notionally but not axiomatically defined (ie that which you set as a subject to examine without itself following from axioms), due to the endless sub-parts one finds and thus cannot have a fully defined basis to stand on.

[John Doe]

1+2+3+4+5+6.... = -1/12

I've been on the wikipage, after reading (and not understanding some of it), isn't there some kind of flawed logic similar to Achilles not being able to catch a turtle? (instead of making a mistake with infinitely small distances, n is infinitely big).

When we write c=1+2+3+... aren't we already making the wrong premise that c is a finite number, and therefor with the wrong start we only can get a wrong answer.

[Kyriakos]

^For the record, there is no "flawed logic" in that paradox (or others) by Zeno, cause his point is exactly that mental human abilities allow for infinite divisions, while senses will cause you to see Achilles run past the tortoise pretty soon. The senses do not factor (at least in this manner, or apparently) infinity, was Zeno's argument, basically, and he (as known by the dialogue 'Parmenides') was just out to back in reduction ad absurdum form the premises of Parmenides, his teacher.

The Eleatic philosophers were of the core view that everything is a Oneness, and thus the sense that we see or experience many distinct things is an illusion. Infinity was one of their central notions when trying to present this view as correct. The idea of "atom" was coined/theorised by Democritos of Abdera, in direct response to the Eleatic philosophies. (also check my last post for Democritos' own issues regarding the notion of the Atom when used in math and not in nature where he intended it for).

[Gaidin]

The thing about math that everybody's gotta remember is that the human brain isn't logical. It's got to be trained in logic. That's why the professors that prove this craziness have usually been at these careers for at least a decade if you count the time it took to get their PhD. So when you see a statement like that that's one line and your first instinct is to go 'bull', remember, there's a lot more lines before it left out as a joke on a forum like this.