One fond phrase used to describe mathematics is "Queen and Handmaiden of Science". Intuitively, I sense this as wonderfully fitting, but what does it mean? Certainly there is a scientific ideal in which mathematics is seen as laying down the laws governing everything -- a sort of regal image, or we can say as "ultimately underlying everything", which might be said to suggest an opposite polarity -- the king or queen doesn't underlie everything, but stands above and orders everything. According to the the "underlying everything" point of view, mathematics pertains to [describes?] (or "is"?) the bricks and mortar out of which everything is made -- the lowest stratum. What science is more mathematical than atomic or subatomic physics?
Perhaps it is a key point to say that the language of mathematics doesn't describe, but is, or creates.
Francis Bacon, I have heard, once said "In order to command nature we must obey nature".
What about psychology? Well, there is a connecting strand, which is
that mathematics seems to occur strictly in the theatre of the mind.
Mathematics has less to do with solid external things -- things outside
of ourselves, than anything I know.
Ha! but if mathematics is "anything" isn't it a thing?
Perhaps the trouble is we have a tendency to thingify everything.
Still, I, at least, have a notion of thing, as distinguished from that which is not a thing (being careful not to say "something" that is not a thing).
In higher mathematics you generally can't even make such an analogy, or picture an experiment involving actual physical objects. One begins with definitions, which are in a way very different from definitions as most people know them. The non-mathematician, is likely, I think, to say that a definition is a description of some thing or class of things that enables one to recognize the thing if one sees it. In higher mathematics, things are defined which no one can ever see -- at any rate, not in the way that we see physical things. And more than that can be said -- namely that we don't even know of a physical analogue of the mathematical entity being described.
This illustrates a sense that we (or at least "I") have, that "objective reality" has something to do with "out-thereness". If you and I close our eyes and go into our own world, there may be little commonality in our experience. I'll be having one imaginary conversation, and picturing things, and/or hearing a song in my head, and you'll be having a different imaginary conversation, etc. But if we each open our eyes and each focus on the same "thing" out there, there will be a point of commonality to our experience. By focusing on the object "out there", we achieve a kind of agreement, and perhaps it is no coincidence that it is called objectivity.
So maybe in a nutshell, mathematics has something to do with the field of possibilities that our minds can roam around in, which is an analogy to something "out there". Somehow our minds, without focusing on things in the "outside world", find ourselves, apparently, in the same field. Is it that this "field" is in both me and you? or maybe it isn't anywhere. Maybe one should call it the field of possibilities of existence. Cute word, existence. So easy to say and so baffling.
So mathematics is, among other things, something that trained human minds can do, and we seem to be grappling with all possibilities of existence when we do it, and, sure enough, we can often turn the visions of mathematics into powerful predictions of how those concrete "out there" things will act.
At least that is the starting point. It sort of takes a stand and says "I'm going to start with my experience -- my thoughts and feelings -- and learn as much as I can about that." If I'm not a solipsist, I also become interested in the thoughts and feelings of others. Now my thoughts, feelings, experiences, ... are sort of inescapable. I'm constantly bathed in them. But I can only imagine yours. There is a huge gap.
So there is a great leap from
* I'm inevitably interested in the world of mind and feeling in which I'm constantly bathed -to-
* Being interested in your mind and feelings, with which I have no such direct link.
So this is why if one were to make a frontal assault on "all knowledge",
* the approach via mathematics, which seems to underlie and pervade all predictable phenomena, OR
* the approach via knowledge of mind, which is my inescapable starting point
seem, to me at least, two of the most natural and promising starting points.
Now this is all leaving aside the question of whether making a "frontal assault on 'all knowledge'" is a good idea, or a totally mad idea.
* * *
It might be fair to say that Socrates took the first long look on record at these two fundamental approaches to Knowledge, and that these two approaches represent two fundamental aspects of Socratic thought.
Socrates was always, implicitly or explicitly, concerned with the mind; and what he accomplished, in his explorations of the mind -- in his attempts to arrive at insight through dialogue, was the discovery of some essential truths -- essences or ideals that seemed quite independent of particular things. He was always a kind of psychologist -- seeking knowledge about the universe through close, subtle, meditative, examination of what he found in his mind, and that of others.
One definition of history, found in my Oxford Universal Dictionary, "now rare, except in 'Natural History'", is "a systematic account (without reference to time) of a set of natural phenomena.". "Natural History" was a phrase applied in the 18th-19th centuries, in most cases where we today would speak of "science". This reflects, I think, the prevalent attitude, or approach, to science of that period. This was the age of classification and taxonomy -- of Linnaeus, and the great project of attempting to categorize all species of living things -- of great entomologists, conchologists, ornithologists (like Audubon)...
At any rate, I think one can say that history, as a general approach to knowledge, is the approach that tries to make something coherent out of all the particulars ... whether particular organisms, or particular happenings. The word "history" tends to be associated with the study of kinds of phenomena that resist universal statements. No historical event is an exact replica of any other. No person is a replica of any other person (including identical twins, who are not inwardly identical, in part because of their different histories).
(I wish I could find my copy of Carl Lotus Becker's The Heavenly City of the 18th Century Philosophers, which had some very interesting things to say about the age in which the "history" paradigm was so prevalent).
This is in marked contrast to physics and chemistry, as we have come to know them, where every object thrown up, or dropped, in a vacuum, follows the same path as every other object; where one ounce of pure alcohol has the same properties as any other ounce of pure alcohol; where mixing 2 parts of one element with 3 parts of another has a predictable result; where one electron has exactly the same properties as any other electron.
In history, and in natural history, one tries to find similarities; to say this bird and that bird are of the same species, although one is slightly larger than the other; to say that any member of such-and-such a species of a given sex will have these organs: A, B, C, ... with these shapes and functions, but there is always variation in details, and in the case of "just plain history" (i.e. not "natural history"), any similarities between events, or peoples lives, are dubious, and refuse to fall into reassuring patterns.
Psychology generally takes a historical (broadly defined) approach in that it must deal ultimately with individual human beings, no two alike. Early (Freudian) psychology also put an extreme emphasis on the therapeutic power of individual coming to an remember and understand forgotten episodes of his/her personal history that played a formative role.
The "understanding" that is sought in this psychoanalytic setting, like the "understanding" one seeks to find by reading a good historical paper, or a biography, is quite different from the history sought in the "hard sciences". It is like the "insight" that may be conveyed by some bit of narrative art -- a novel, or story (with or without an explicit moral), or a movie. Generally, you do not derive general "laws" about how the world operates from history, or from stories, or if you do derive such laws you tend to be unaware that you're doing so (this is getting a bit murky). Often we have a sense of having received some "light" from a story or history, but would be unable to say "The lesson I learned is ...". Sometimes we want to go out and retell the story.