Yes, I know, but information gets lost in the process of simplification. This is what makes math useful and powerful: loss of data allows for a skeleton to be revealed.
But metaphors include _both_ the simplification and _also_ the differences:
Example: Where I say [a group of people] is [a plague upon society] I’m making a metaphor.
There are similarities and differences that are expressed together.
In mathematics, it’s still metaphors but they’re agnostic metaphors: the similarities remain, the differences removed. What’s left is quantity or an abstraction of quantity.
Mathematics _does_ have a unique grammar that allows it to express things that are difficult to explain in other ways outside of itself.
And yet, it does.
Example: You express impossible objects in mathematics.
But: Language expresses impossible objects and impossible situations regularly.
It’s embedded in the language, so like a fish can’t see the water, we can’t see it until it’s pointed out.
Example: Time. We speak of time in a very flexible manner: “would have done”, “should have done”, “could have done” are three examples of fictional pasts with alternative timelines and different intentions for each.
These are IMPOSSIBLE things. They didn’t happen. But we express them, talk about them, and think about them as *if* they’re real.
Back to math: When I say “quantity” I’m not just talking about 1 2 or 3 here.
What I mean is: At its root, math resolves to quantity. Quantity of dimensions. Quantity of [label].
You can tell by the things that mathematics fails at. It fails at 0. It fails at infinity. They can still be useful, but if there’s no quantity, or an incalcuable quantity, it fails.
Yet, it handles imaginary quantities just fine, and with algebra, it can carry unknown quantities freely: as long as there is a quantity there and not an infinity or a zero, at which point, it fails again. It must have quantity.
Yes and mathematics-as-metaphor doesn’t mean it always corresponds to reality: it’s metaphors to itself.
Here’s the connotation that math carries and there’s more than this:
When using math, the connotation it carries is that “there are other things that MAY BE attached to this that do not matter. They can be ignored”.
Example: When people are treated as numbers, people have qualities that math can safely ignore. Many calculations can be made involving people because they are no longer people, they are numbers.
Comparisons can be made, and run through various formulas that are akin to folding and ripping paper, shredding it into pieces, reconstructing it into new forms, and what you end up with is conclusions.
These conclusions are then taken from their agnostic form and the people are returned to the numbers, ideally.
Yet, the loss of information in the process, not critical in math, _might_ be critical to that which it represents.
I’ll respond to your next statement in a sec
Is all abstraction metaphorical? That’s a good question. I believe it must necessarily be but I have nothing to back that up with yet.
Ok: the answer is yes. Example:
You speak of abstraction. This is “as opposed to concrete”.
Already, you are using metaphors. It is difficult to escape them.
Metaphors are always abstractions and yet the “what is an abstraction” runs rather deep.
If you pick up a ball, pick up another ball, you can say: I have ‘this’ and I have ‘that’.
Are the objects the same? No. They’re different.
But our brains are limited in capacity and ability. We have to simplify.
So, we end up with a concept of “ball” by comparing similarities and they have enough in common to form a template of “ball”.
So, you have a quandry: These look _similar enough_ how can express where they lack uniqueness?
You have quantity of ball. Two balls.
This ability goes DEEP into our evolutionary past it seems: all the way back to bacteria, they count in some form. They know self, other, self vs other, what self-and-other have in common, at least behaviorally. [we analyze their chemical communication and behavioral patterns].
So, the ability to form abstractions is there from the start. They’re simplifications.
Oh absolutely. But even in “regular language”, impossible things are just as everyday as the 196,883 dimensional object is in mathematics.
We’re just used to is so we don’t notice it.
Language goes far beyond Aristotelian, 3D objects just as mathematics does.
They’re both descriptive, both of things that are real and things that are impossible to express any other way.
And, there are things impossible to describe in math *or* spoken/written language.
Interestingly though: the object with 196883 dimensions? It was described using “regular language”. It was communicated and we understood it to some degree.
There’s something to be said about that facility.
I have _yet_ to come across some method of description that wasn’t metaphorical. This is why I hold (at present) metaphor to be at the basis. I can’t see any way to describe anything in _any_ capacity without metaphor: and I see mathematics as a special case of metaphor that is powerful in its own right.
But – are you really seeing land? What is it to see? You receive impressions because you’re paying attention to them. Your attention ‘looks out’ and brings into you a 2D image (which also isn’t really 2D – nothing is really 2D) – upon your retina and goes into your brain.
This image is processed in your mind through a series of templates, activating your grammar system and overall comprehension until you are “seeing land”.
But within the brain, the “thing seen” only is seen BECAUSE internally the brain uses metaphors to itself: “this is like that”. Comparisons of intangibles – abstractions.
But even the distinction between literal and metaphor is one for our convenience. In one sense, it can be said that “all is metaphor”. In another sense, it can be said that “all is literal / concrete”.
But if you were to take the whole set of literal + abstract together to form a whole picture of everything, the literal is but a small subset of abstraction: the literal is constructed entirely from abstraction, metaphors built upon metaphors to aid in our understanding, forming such complexity that we don’t know it’s there at all. But it is.
Metaphor is not mere. Metaphor is at the core of mathematics, right down to its beginnings.
We use shorthand for it now, but go back to its roots, the “long hand” version of math *is* based upon tangibles.
The tangibles came first. The abstraction came later on. Effective shorthand, but shorthand-for-something just the same.
The shorthand is functional enough to act as if it is a machine, able to produce things that are greater than the parts they came from.
But you have to respect its roots.
Assumptions on my part:
Everything we know is processed through a human mind.
The human mind compresses data into stereotypes / templates / seeks patterns. It’s what we do without even trying. We _have_ to.
This stuff is SO EMBEDDED in our language that’s nearly impossible to escape the use of metaphor to describe things.
I’m not speaking about “Metaphors as describe in English class” or somebody’s dictionary entry.
Rather, I’m speaking of the nature of metaphor itself.
You can dismiss my argument if you like based upon dictionary distinctions between abstraction and metaphor if you like. Fact is, what I’m referring to as metaphor *isn’t* metaphor in the sense you’re using it, which is making communication difficult.
If it makes comprehension *easier for you* to substitute abstraction for metaphor – if it helps your befuddlement, then go for it. It’s the underlying concept I’m trying to get across that matters here.
Indeed – and the Platonic triangle functions “as if real” for us humans because our cognitive systems automatically simplify.
Plato’s forms conform nicely to how our brains compress information received. The forms don’t seem to correspond to “reality as measured” but they work well with our simple cognitive systems as does the mathematics based upon the primitive geometry with the axioms and proofs and all that was built upon it since then.
We build machines out of it. It’s amazing stuff. We run into problems when engineering because reality doesn’t conform to geometric/mathematical perfections but that’s why engineering has tolerances: the math can’t be perfect.
They’re agnostic metaphors. They don’t know what they refer to. Numbers are about ignorance of all the facts except for that which is necessary in the situation where the numbers are used.
Yes, within language, there is an amazing amount of flexibility of description. It doesn’t necessarily have to describe things “as they are” but we can form “conceptual buckets” to assist us in separating a “this” from a “that”.
“Literal” vs “abstract” for example.
We can say “These things go into the ‘literal bucket’. There is a ‘something’ that all of the things in the literal bucket have in common. They form the “concept of literal”.
Same for the “abstract bucket”. There is a ‘something’ in all the things we put in the ‘abstract bucket’ that form the ‘concept of abstract’.
Yet, sometimes you get things that can go into either bucket. Or, neither. Or have to switch sides depending on the context.
We have tools to make it easier on us that works in many situations. Dictionaries for example. Cooperation in communication is extremely useful.
Of course there’s no buckets and yet the way we use language, there might as well be.