Dace wrote:
>
>. If
> equations could exist in human brains, then what's to stop them from
> existing on pieces of paper? In other words, you're still taking these
> things at face value, which means you're conceding that they're eternal and
> ideal. We must never give an inch to Plato! Equations have no
> self-existence. They do not exist on paper or in brains. They are abstract
> and exist only to the extent that we imagine them. You only *imagine* that
> the equation is written down on the piece of paper. In fact, when you look
> at the paper, what your eye sees is just bits of ink. It's the *mind*, not
> the eye, that "sees" the equation. In reality, the equation does not exist
> except in your mind.
I suppose math teachers do not write equations on the board, or project them on screens etc. etc. Things written on paper are not eternal nor are things in brains. In a sense, however, equations are obviously not destroyed when a mathematician dies or a piece of paper with an equation on it is burned. I suppose this is part of what is meant by saying that equations are abstract. Numbers are abstract too in that once the kid learns them you can burn up the number-picture cards. This does not mean I cannot type 5 + 3 = 8. You can read that on your screen and recognise that it is true. It is not that your mind does something special independently of your eyes. YOU recognise it is true, and if you were blind you would not recognise it was true (unless there were some "translation" into braille or aural format.)
You do not imagine you write an equation on a piece of paper when you do it. Minds don't see anything except in a metaphorical sense. Your eye does not see just bits of ink rather YOU see an equation. Someone who does not recognize equations may just see bits of inks.
There is a difference between imagining you are seeing an equation written on a piece of paper and seeing an equation written down on a piece of paper. Except in special circumstances it would be most odd to say "I imagine I see an equation written on that piece of paper" when you are actually looking at an equation written on the paper. You might say this when there are ink blots on the paper that look as if they might be letters and symbols but are so misshapen or mixed up with other shapes that it could be anything, something like a Rohrschach test pattern.
Of course you put scare quotes around "imagine", "see","mind" so I suppose all this will be regarded as irrelevant. But unless you explain what special meaning you give to the terms in quotes I haven't got a clue as to what you are talking about.
According to you equations exist only in minds. I have a few questions.
1) Take the equation x + y = 10. Is there a different equation in your mnd and my mind and everyone else's mind who reads this? Are there then equations or an equation we are talking about?
2) How do I know that the equation in your mind is the same as in mine? I cannot access your mind. Ditto everyone else who reads the equation.
3) Is the same equation in everyone's mind? And how are we to talk about the equation unless we assume this? But each "imagination" or "idea" corresponding to what was written down will be distinct as in each person's mind. How can this be unless there is something called the equation and these ideas are but instances or instantiations of the same equation in different minds. Of course this gets us back to something like Platonism.
4) If the mind sees the equation this seems to imply that the equation is something distinct from the mind just as when we see something it is not typically the eye except as reflected in some external medium such as a mirror. The implication of using "see" is that the equation itself is not within the mind.
You seem to raise more metaphysical problems than you solve. That's what comes of talking funny! Personally, I think there are equations all over the place: stored on disks and hard drives, on pages in books, in people's brains, etc.etc.
Cheers, Ken Hanly