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...
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...
In a way (but this is not a pure parallel: ) it seems to be similar to claiming that while cave-paintings by prehistoric people were 'art', future art (eg in 2014) is not really tied to creativity cause it is not 'primitive'. But it seems to me to be a fault of viewing the crucial parameters here. Current math is not independent of classical/hellenistic era math. Most of it is even directly tied to those (eg Gauss' work on infinite series and Eratosthenes, or my afforementioned allusion to proto-calculus by Archimedes, or examinations of hellenistic plane geometry by Fermat (tried to reproduce the entire hellenistic text of De Locis Planis), etc). 
