I thought there would be no harm posting this here too... I kept the full post, cause why not? 
We have discussed this here (at CFC) in the past as well; maybe there can be a new discussion?
Preface
I do fear that perhaps this post of mine (my fourth here) may cause a few negative reactions. I do try to approach this from a philosophical viewpoint, as befits my studies. It goes without saying that I may be wrong, and would very much like to read your views and even more so any reasons that my own position may be identified as untenable. I can only assure you that to me it currently seems that mathematics are not cosmic but anthropic.
*
There are so many quotes about mathematics, from celebrated mathematicians, philosophers, even artists; some are witty yet too polemical to identify as useful in a treatise that aspires to discuss whether math is merely anthropic or cosmic, and others are perhaps too focused on the order itself and thus come across a bit like the expected fawning of an admirer to his or her muse.
Yet the question regarding math being only a human concept, or something which is actually cosmic, is an important one, and it does deserve honest examination. I will try to present a few of my own thoughts on this subject, hoping that they may be of use – even if their use is simply to allow for fruitful reflection and possible dismissal.
It is evident that mathematics have value. It is also evident that they allow for technological development. They do serve as a foundation for scientific orders that rest on experiment and thus are invaluable. However we should also consider what the primary difference between math as an order and scientific orders (physics, chemistry etc) easily let’s us know about math itself:
Primarily math differs from science in that it secures that its results are valid not from experiment, data and observation, but axiom-based proof. The use of proof in math is often attributed to the first Greek mathematicians, and specifically to either the first Philosopher, Thales of Miletus, or his students, Anaximander and Pythagoras. Euclid argued that the first Theorem that math presents is the one by Thales, which has to do with analogies between parts of 2D forms (eg triangles) inscribed in a circle. The idea of a proof proceeding from axioms, of a Theorem, is fundamental in mathematics – and it also is a crucial difference between math and orders such as physics. Fields of science that have to do with observing (and interacting with) the external world do significantly differ from a field (math) which only requires reflecting on axiomatic systems.
Given the above is true, it does follow that a human is far more connected to math than to any study of external objects: they are tied to math without even trying to be tied to it, given math exists as a mental creation and not one which requires the senses to intervene.
But what does “being more connected” mean, in this context? Is math actually intertwined with human thought of all kinds? Obviously we do not innately know about basic “realities” of the external world, such as weight and impact; the risk of a free-fall is something that an infant has to first accept as a reality without grasping why it is so. On the contrary we do, by necessity, already have fundamental awareness of the (arguably) most basic notion in all of mathematics: the notion of the monad.
The monad is the idea of “one”. That anything distinct is a “one”, regardless of whether we mean to include it in a larger group or divide it to constituent parts: each of those larger groups are also “one”, and the same is true for any divisions. “Oneness”, therefore, as the pre-socratics already argued (and Plato examined in hundreds of pages) is arguably one of the most characteristic human notions, and a notion which is generally inescapable and ubiquitous. “One” is also the first digit and the meter of the set of natural numbers (1,2,3,4…), and this is because the human mind fundamentally identifies differences as distinct, even when the difference may become (in advanced math) extremely complicated and of peculiar types. Yet the humble set of natural numbers also gives us an interesting sequence when altered a bit: the so-called Fibonacci sequence, which I think is a good example to use so as to show why I think that math are only human and not cosmic.
The Fibonacci sequence progresses in a very specific way: each part is formed by adding the two previous parts. The sequence begins with 1 (or 0 and 1), so the first parts of it are (0), 1, 1, 2, 3, 5, 8,13. The entire sequence diverges from both sides (alternating between the next part presenting a numerical difference just smaller or just larger) to the golden ratio, and forms a pretty spiral form (wiki image: https://en.wikipedia.org/wiki/Fibonacci_number#/media/File:FibonacciSpiral.svg). Yet for me it is of more interest that humans do happen to observe a good approximation of this specific, mathematical spiral, on some external objects; namely the shells of a few small animals.
It is pretty clear that the shell of some external being is not itself aware of mathematics. One could argue, of course, that “nature” itself is filled with mathematics, and thus in some way a few external forms happen to approximate a specific spiral, and the tie to the golden ratio etc is only to be expected given nature (and by extension, perhaps, the Cosmos itself) is mathematical. Certainly this can appear to provide an answer; or to be precise it would at least present a cause for this appearance of mathematics and of a specific spiral in the external world. Is it really a good answer, though? In other words, do we observe the Fibonacci or golden ratio spiral approximation on the external world because the external world itself is tied to math, or do we do so because we are tied to math in an even deeper way than we realize and could only project what we have inside of our mental world onto anything external?
My view is that humans are so bound to math (regardless of how knowledgeable one is in mathematics) that we cannot but view the world mathematically. Rockets are built, using math, and by them we can even leave the orbit of our planet – yet consider whether what allowed us to realize how to achieve so impressive a result was not math alone, but math as a kind of very anthropic cane or leg by which we slowly learned to move about:
In essence I do think that due to the human species being so obstructed from developing far more advanced mathematics (to put it another way: due to how difficult advancing math can be even for the best mathematicians) we tend to not identify that math itself is not the cause of development, not the cause of movement and progression, but a leg - the only leg - we have to familiarize ourselves with because we aspire to move on this plane. Imagine a dog which wanted to move from A to B, but couldn’t use its legs. At some point it manages to move one of them, and then enough so as to finally get to B. It is undoubtedly a major achievement for the dog. But the dog shouldn’t proceed to claim that the dirt between A and B is made of moving legs – let alone that it is the case for the entire Cosmos.
I only meant to briefly present my thoughts on this subject, and wish to specify (what very likely is already clear to more mathematically-oriented readers of this post) that my personal knowledge of mathematics is quite basic. I approach the subject from a philosophical and epistemological viewpoint, which is more fitting to my own University studies (Philosophy).
We have discussed this here (at CFC) in the past as well; maybe there can be a new discussion?
Preface
I do fear that perhaps this post of mine (my fourth here) may cause a few negative reactions. I do try to approach this from a philosophical viewpoint, as befits my studies. It goes without saying that I may be wrong, and would very much like to read your views and even more so any reasons that my own position may be identified as untenable. I can only assure you that to me it currently seems that mathematics are not cosmic but anthropic.
*
There are so many quotes about mathematics, from celebrated mathematicians, philosophers, even artists; some are witty yet too polemical to identify as useful in a treatise that aspires to discuss whether math is merely anthropic or cosmic, and others are perhaps too focused on the order itself and thus come across a bit like the expected fawning of an admirer to his or her muse.
Yet the question regarding math being only a human concept, or something which is actually cosmic, is an important one, and it does deserve honest examination. I will try to present a few of my own thoughts on this subject, hoping that they may be of use – even if their use is simply to allow for fruitful reflection and possible dismissal.
It is evident that mathematics have value. It is also evident that they allow for technological development. They do serve as a foundation for scientific orders that rest on experiment and thus are invaluable. However we should also consider what the primary difference between math as an order and scientific orders (physics, chemistry etc) easily let’s us know about math itself:
Primarily math differs from science in that it secures that its results are valid not from experiment, data and observation, but axiom-based proof. The use of proof in math is often attributed to the first Greek mathematicians, and specifically to either the first Philosopher, Thales of Miletus, or his students, Anaximander and Pythagoras. Euclid argued that the first Theorem that math presents is the one by Thales, which has to do with analogies between parts of 2D forms (eg triangles) inscribed in a circle. The idea of a proof proceeding from axioms, of a Theorem, is fundamental in mathematics – and it also is a crucial difference between math and orders such as physics. Fields of science that have to do with observing (and interacting with) the external world do significantly differ from a field (math) which only requires reflecting on axiomatic systems.
Given the above is true, it does follow that a human is far more connected to math than to any study of external objects: they are tied to math without even trying to be tied to it, given math exists as a mental creation and not one which requires the senses to intervene.
But what does “being more connected” mean, in this context? Is math actually intertwined with human thought of all kinds? Obviously we do not innately know about basic “realities” of the external world, such as weight and impact; the risk of a free-fall is something that an infant has to first accept as a reality without grasping why it is so. On the contrary we do, by necessity, already have fundamental awareness of the (arguably) most basic notion in all of mathematics: the notion of the monad.
The monad is the idea of “one”. That anything distinct is a “one”, regardless of whether we mean to include it in a larger group or divide it to constituent parts: each of those larger groups are also “one”, and the same is true for any divisions. “Oneness”, therefore, as the pre-socratics already argued (and Plato examined in hundreds of pages) is arguably one of the most characteristic human notions, and a notion which is generally inescapable and ubiquitous. “One” is also the first digit and the meter of the set of natural numbers (1,2,3,4…), and this is because the human mind fundamentally identifies differences as distinct, even when the difference may become (in advanced math) extremely complicated and of peculiar types. Yet the humble set of natural numbers also gives us an interesting sequence when altered a bit: the so-called Fibonacci sequence, which I think is a good example to use so as to show why I think that math are only human and not cosmic.
The Fibonacci sequence progresses in a very specific way: each part is formed by adding the two previous parts. The sequence begins with 1 (or 0 and 1), so the first parts of it are (0), 1, 1, 2, 3, 5, 8,13. The entire sequence diverges from both sides (alternating between the next part presenting a numerical difference just smaller or just larger) to the golden ratio, and forms a pretty spiral form (wiki image: https://en.wikipedia.org/wiki/Fibonacci_number#/media/File:FibonacciSpiral.svg). Yet for me it is of more interest that humans do happen to observe a good approximation of this specific, mathematical spiral, on some external objects; namely the shells of a few small animals.
It is pretty clear that the shell of some external being is not itself aware of mathematics. One could argue, of course, that “nature” itself is filled with mathematics, and thus in some way a few external forms happen to approximate a specific spiral, and the tie to the golden ratio etc is only to be expected given nature (and by extension, perhaps, the Cosmos itself) is mathematical. Certainly this can appear to provide an answer; or to be precise it would at least present a cause for this appearance of mathematics and of a specific spiral in the external world. Is it really a good answer, though? In other words, do we observe the Fibonacci or golden ratio spiral approximation on the external world because the external world itself is tied to math, or do we do so because we are tied to math in an even deeper way than we realize and could only project what we have inside of our mental world onto anything external?
My view is that humans are so bound to math (regardless of how knowledgeable one is in mathematics) that we cannot but view the world mathematically. Rockets are built, using math, and by them we can even leave the orbit of our planet – yet consider whether what allowed us to realize how to achieve so impressive a result was not math alone, but math as a kind of very anthropic cane or leg by which we slowly learned to move about:
In essence I do think that due to the human species being so obstructed from developing far more advanced mathematics (to put it another way: due to how difficult advancing math can be even for the best mathematicians) we tend to not identify that math itself is not the cause of development, not the cause of movement and progression, but a leg - the only leg - we have to familiarize ourselves with because we aspire to move on this plane. Imagine a dog which wanted to move from A to B, but couldn’t use its legs. At some point it manages to move one of them, and then enough so as to finally get to B. It is undoubtedly a major achievement for the dog. But the dog shouldn’t proceed to claim that the dirt between A and B is made of moving legs – let alone that it is the case for the entire Cosmos.
I only meant to briefly present my thoughts on this subject, and wish to specify (what very likely is already clear to more mathematically-oriented readers of this post) that my personal knowledge of mathematics is quite basic. I approach the subject from a philosophical and epistemological viewpoint, which is more fitting to my own University studies (Philosophy).
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