Image Credit & Copyright: Optical (RGB+Ha): Aldo Mottino & Ezequiel Bellocchio (Argentina); Infrared: ESO/J. Emerson/VISTA.
I like to compare mathematical concepts to an Arch(so did Jacob Bronowski; but, I think I've figured out the true comparison between the two). The arch represents the unification of concepts, the delicate balance between too little and too much that all mathematical concepts are. The arch also can't be built like a crossbeam(two vertical columns with a crossbeam laid on top haphazardly). The post and lintel is constructed from current perspective viewpoint. Kind of like what our senses initially say about the Earth - that it is flat. It takes looking behind the curtain, figuratively speaking, to see that the Earth is round and it goes around the sun and not the other way. The arch takes insight. Those insights are often underlying structural analogies.
All mathematical concepts are abstractions. Abstractions are the general form that many structures can take on. Take a given relation, and many different content can make sense with that relation. I love you; you love me; two apples and two oranges make up the number two. One could say the arch is composed of a verb, the keystone, and the contents that make sense with that keystone; each elemnt of the arch is differnt in some ways(some being higher or lower on the arch), but they have a similar relation with respect to the keystone.
The elements of the arch that make sense with respect to the keystone is similar once again to the way the various subject and object nouns of a verb need to make sense. This making sense with respect to a given verb(see Suzanne K Langer's "Introduction to Symbolic Logic" for a good discussion of how symbolic logic relates to the abstract nature of mathematics) brings up something else I think I've found about the mathematical activity.
I've been watching James Burke's "Connections" and "The Day the Universe Changed" a lot, and I can't help noticing certain technological groupings. James Burke tries to show the connections between science and all aspects of the human condition through history. Some connections are just kind of outside influences things. But, some connections between the developments of technologies seem to me to have a far more profound 'connections.' Like in episode one of Connections(the first series is the only one i've seen), he shows how agriculture led to a host of technologies. He shows how agriculture leads to the specialization of jobs; the farmers do the farming allowing others to do government, military duty, maybe further down the road some can do science freely. I point out that 'clearing the fields' suddenly makes mankind need to take on the jobs of irrigation, fertilization, and pest control. These concepts 'make sense' in relation to the verb/act of 'clearing of the fields.' James Burke mentions other possible related technologies that makes sense with respect to agricultural civilization - like storage containers for the surplus grain. In episode three, he goes into how the horse changed the economies and created a bunch of associated technologies just like agriculture did - the horseshoe, the sturip, the metallergy of the knight's armour. In episode two, he mentions how the vacuum leads to noting how mice suffocate in it; flames go out, bells don't ring in vacuums; and he notes that with the vacuum, following the history after can take many different paths - "we could go from the vacuum pump to the investigation of air to the discovery of oxygen, or vacuum pump to steam engine, or to the cathode ray tube and so on. See, by 'clearing of the fields' so to speak, certain concepts just come about; they make sense with respect to daming nature up so to speak. I associate this daming up of nature(creating a water dam 'generates' electricity) with idealization in mathematics. Idealization leads to the definitions in axiomatic mathematics. This all probably came to my mind because I've read Jacob Bronowski's "Origins of Knowledge and Imagination".
In Jacob Bronowski's "Origins of Knowledge and Imagination", he points out that we figure out the universe from our current perspective; and, our current perspective is always finite while the universe is this infinitely connected whole. To get at nature, we have to introduce an artificiality(an idealization/ in mathematics, a definition); this definition creates a vacuum in nature that we need to introduce the keystone and the elements of the arch to keep that vacuum from collapsing back in on us again. Jacob Bronowski argues that when we first evolved from our non-human selves, we were one with nature; our language was command sentences. And that we decodes the words from the sentence by a process of generalization and specialization. His ideas are nice, but until I watched James Burke's "Connections and The Day the Universe Changed" enough, and noticed these 'clearing of the fields' and how fundamental the vacuum has been to the investigation of new concepts and technologies, even I sometimes wondered how seriously to take his little book.
For a quick mathematical example; introduce a line on to two parallel lines; what makes sense is that opposite angles are equal. It's pure language. Physicists love to point out Romer's discovery of the finite speed of light. He observed the motions of the Galilean moons and couldn't help notice an inconsistency in the times of the moons motions; the only thing that made sense was that the light must have a finite speed of light; establishing a boundary creates problems; the establishing of a boundary asks what is the opposite shape of the boundary? The Newtonian laws had a certain boundary established; Romer applied them to the motions of the moons of Jupiter and found a problem; the opposite shape of the consistent boundary established meant the finite speed of light.
Point is that a certain noun words 'work' or 'make sense' with a given verb. Similarly, the finite speed of light 'made sense' with respect to newtonian mechanics; a consistent 'daming up of nature.'
More mathematical examples would be how the irrational numbers were derived out of a problem of the pythagorean theorem; take the case where the sides equal one and bring down the hypotenuse; this line cannot be measured! Similarly, negative numbers, imaginary numbers in quadratic equations, group and field theory in Galois solution(or disproof) of the general equation of the fifth degree all 'spring out' of daming up nature in just the right way.
In logical proof, you split up the conjecture in terms of hypotheses and conclusion; you know your rules of how to transform one side of another to get things on one side and you find the keystone the bridges the vacuum made up.
Einstein's theories of Relativty, Quantum theory, and Chaos theory didn't throw away Newtonian theory; they generalized it and derived it. E.T. Bell notes the Hilbert problem solved by Dehn which generalizes the Egyptian solution to the volume of the truncated pyramid. E.T. Bell stresses these phenomenon of abstraction and generalization throughout the history of mathematics.
Mathematics is arches within arches. It is a cathedral of the mind alright. I feel like I've been through a cathedral when I read E.T. Bell's "The Development of Mathematics." E.T. Bell shows the unity and extent of mathematics of all mathematics; he also shows the connections between modern mathematics and the great accomplishments of mathematics of as far back as he could get when he wrote this book.
There's been great artists of various arts; but, outside of passing down to future generations new techniques, a given artist creates their art, and the next creates their art(in the blues, they build up on one anothers songs actualy!). But, in mathematics each generation opens up the previous generations mathematics, generalizes and finds new unities. Mathematics is by far now the deepest human construct. It's possiblities are infinit(as proved by Kurt Godel; Godel's theormes say an inconsistent finite set of axioms can prove an infinity of truths; a consistent set of finite axioms cannot prove an infinity of truths; mathematicians pick the second possibility; hence their activity will always continue).
Jacob Bronowski essentialy argues that humanity is the science and technologicaly dependent species. We came from ignorance. Mathematics is the ultimate expression of our effort to comprehend the universe. E.T. Bell shows this development from as early times as he could when he wrote it. He shows the connections between all the great accomplishments of the past and all the latest mathematics. It's kindof remarkable how a study of mathematics shows the main outlines of human history!