Who will allow you to multiple your debt with another’s debt to get rid of it?
2 −1 + −2 − 1 = (2 − 2) −1 = 0 −1 = 0
But...
2 −1 + −2 −1 = −2 + −2 * −1 = 0
Therefore...
-2 * −1 = 2
Ian Stewart, Professor Stewart’s Cabinet of Mathematical Curiosities, Profile Books, 2008, pages 37-38;
So mathematics is mentally-created, it looks objective because of primordial choices we have made? As a form of a subconscious of the Species and we’ve created computers because we think that way and choose to think that way? Our truths may be grounded on habit and rationality may be a self-imposed restriction, nature being more non-bivalent than we would like it to be. Under the same logic, the furniture in our room might have been subject to similar primordial choices. Oh, and especially I, Ourselves, Identity.
The degrees of what’s available in nature might be infinitely greater than we think and a self-imposed boundary might be inherently limiting.
Hofstadter opens with the story of J.S. Bach’s Musical Offering for King Frederick, which contains a particular canon that sneakily shifts from one key to another before its apparent conclusion, and when this modulation is repeated 6 times, the piece ends up at the original key but one octave higher. This is our first example of a “Strange Loop”:
The “Strange Loop” phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some heirarchical system, we unexpectedly find ourselves right back where we started. (Here, the system is that of musical keys.)
Other examples occur in the drawings of M.C. Escher, for example this famous one.
Units would help a lot in your debt example. 2 $-1 = $2, two instances of $1 debt is $2 of debt. The multiplier and multiplicands are DIFFERENT—the multiplicand is dollars of debt and the multiplier is a count of debts. And −2 $-1 = $2, yes. If you have negative two debts of $1 (that is, you remove two of them), you have a net of positive $2.
Never do you multiply a debt by another debt—that would give you square dollars, which makes no sense.
I think you meant (-2 * −1 = $2)
I meant, multiply by a negative count of debt and not itself. So a debt multiplied by a negative count of debts leads to no debt at all, a positive. I’m not sure how you can have a negative count of debts.
$2 debt squared does make sense, though, it is $4 and no debt. So by our mathematics, I could call the bank and ask them to multiply my debt with yours, I would return a positive.
The point I am making is that we’ve made it this way because have chosen to. It says to me that mathematics is more of a mental creation, albeit a very useful one and that nature might be infinitely greater than our own self-imposed boundaries.
Take a look at this picture does the water goes up or down? Is bivalent thinking necessarily nature or simply a mental creation? When it comes to truths (true or false) or computers (by primordial decisions) (1 or 0)
$2 debt squared does make sense, though, it is $4 and no debt.
No, it is $$4.
If that’s what you meant to write, and it’s also obvious to you that you could have written 40000¢¢ instead and still been completely accurate, then I’d love to know if you have any ideas of how this computation could map to anything in the real world. I would have thought that “kilogram meters squared per second cubed” was utter nonsense if anyone had just tried to show me the arithmetic without explaining what it really meant.
If that’s not what you meant to write, or if it takes a second to figure out why $$4 isn’t 400¢¢ instead of 40000¢¢, then you’ve just got the illusion of sense going on. And yes, I just noticed that pun and it wasn’t intentional.
A negative count maps well into my understanding of physical reality. Positive count is adding multiple similar things, negative count is removing those things. Removing 2 debts is the same as adding 2 of the values of the debts. Thus −2 * $-2 = $4. Removing two $2 debts is equal to adding $4.
There are elements of math and symbolic reasoning that don’t map to reality, fine. But those parts which DO map well, are pretty strong, and are empirically correct in addition to being symbolically/conventionally well-formed. Mathematics is a mental creation, but that doesn’t make it unrelated to reality—it’s a pretty good and well-tested model of our universe.
As to that picture, the water goes neither up nor down—it’s a still drawing.
I’m sorry if I don’t understand, but multiplying my debt with a greater debt leads to no debt. It is true as the mathematics show. If we say to the bank, check account A debt and multiply with account B debt, account A will have no debt. It is independent on how you want to phrase it.
What does each operation in your equation represent? “Removing two $2 debts is equal to adding $4”
It is true because the mathematics has to stay consistent, it is based on primordial choices. That’s my point, we choose it this way.
There are elements of math and symbolic reasoning that don’t map to reality, fine. But those parts which DO map well, are pretty strong, and are empirically correct in addition to being symbolically/conventionally well-formed. Mathematics is a mental creation, but that doesn’t make it unrelated to reality—it’s a pretty good and well-tested model of our universe.
As to that picture, the water goes neither up nor down—it’s a still drawing.
So, if I understand right, you think that mathematics is a mental creation and does map well to reality, but it doesn’t make it unrelated to reality. Reality seems to be independent of our maps, and a relation between a map and it would be a mental one. Yet reality is beyond any maps or limits.
Well, the drawing is non-bivalent yet we choose our thinking to be such, as evident towards a lot.
multiplying my debt with a greater debt leads to no debt
You keep ignoring the issue of units. Multiplying dollars by dollars would lead to square dollars, which is a mistake (and not just because dollars are actually rectangular in shape). It is “-2 × $-2”, not “$-2 × $-2″. Money are not multiplied by money. Money are mutliplied by… number of accounts, or number of repetitions, or other dimensionless numbers.
Consistency in Arithmetic
Double the debt: 2 −1 = −2 *Ok
But: −2 −1 = 2 *Ok?
Who will allow you to multiple your debt with another’s debt to get rid of it?
Ian Stewart, Professor Stewart’s Cabinet of Mathematical Curiosities, Profile Books, 2008, pages 37-38;
So mathematics is mentally-created, it looks objective because of primordial choices we have made? As a form of a subconscious of the Species and we’ve created computers because we think that way and choose to think that way? Our truths may be grounded on habit and rationality may be a self-imposed restriction, nature being more non-bivalent than we would like it to be. Under the same logic, the furniture in our room might have been subject to similar primordial choices. Oh, and especially I, Ourselves, Identity.
The degrees of what’s available in nature might be infinitely greater than we think and a self-imposed boundary might be inherently limiting.
A Musico-Logical Offering
Is the water going up or down?
Units would help a lot in your debt example. 2 $-1 = $2, two instances of $1 debt is $2 of debt. The multiplier and multiplicands are DIFFERENT—the multiplicand is dollars of debt and the multiplier is a count of debts. And −2 $-1 = $2, yes. If you have negative two debts of $1 (that is, you remove two of them), you have a net of positive $2.
Never do you multiply a debt by another debt—that would give you square dollars, which makes no sense.
edit: fix typo
Actually, I would argue that multiplying by −1 is just the notation picked to express this particular operation which is flipping the sign.
I think you meant (-2 * −1 = $2) I meant, multiply by a negative count of debt and not itself. So a debt multiplied by a negative count of debts leads to no debt at all, a positive. I’m not sure how you can have a negative count of debts.
$2 debt squared does make sense, though, it is $4 and no debt. So by our mathematics, I could call the bank and ask them to multiply my debt with yours, I would return a positive.
The point I am making is that we’ve made it this way because have chosen to. It says to me that mathematics is more of a mental creation, albeit a very useful one and that nature might be infinitely greater than our own self-imposed boundaries.
Take a look at this picture does the water goes up or down? Is bivalent thinking necessarily nature or simply a mental creation? When it comes to truths (true or false) or computers (by primordial decisions) (1 or 0)
No, it is $$4.
If that’s what you meant to write, and it’s also obvious to you that you could have written 40000¢¢ instead and still been completely accurate, then I’d love to know if you have any ideas of how this computation could map to anything in the real world. I would have thought that “kilogram meters squared per second cubed” was utter nonsense if anyone had just tried to show me the arithmetic without explaining what it really meant.
If that’s not what you meant to write, or if it takes a second to figure out why $$4 isn’t 400¢¢ instead of 40000¢¢, then you’ve just got the illusion of sense going on. And yes, I just noticed that pun and it wasn’t intentional.
thanks, corrected.
A negative count maps well into my understanding of physical reality. Positive count is adding multiple similar things, negative count is removing those things. Removing 2 debts is the same as adding 2 of the values of the debts. Thus −2 * $-2 = $4. Removing two $2 debts is equal to adding $4.
There are elements of math and symbolic reasoning that don’t map to reality, fine. But those parts which DO map well, are pretty strong, and are empirically correct in addition to being symbolically/conventionally well-formed. Mathematics is a mental creation, but that doesn’t make it unrelated to reality—it’s a pretty good and well-tested model of our universe.
As to that picture, the water goes neither up nor down—it’s a still drawing.
I’m sorry if I don’t understand, but multiplying my debt with a greater debt leads to no debt. It is true as the mathematics show. If we say to the bank, check account A debt and multiply with account B debt, account A will have no debt. It is independent on how you want to phrase it.
What does each operation in your equation represent? “Removing two $2 debts is equal to adding $4”
It is true because the mathematics has to stay consistent, it is based on primordial choices. That’s my point, we choose it this way.
So, if I understand right, you think that mathematics is a mental creation and does map well to reality, but it doesn’t make it unrelated to reality. Reality seems to be independent of our maps, and a relation between a map and it would be a mental one. Yet reality is beyond any maps or limits.
Well, the drawing is non-bivalent yet we choose our thinking to be such, as evident towards a lot.
You keep ignoring the issue of units. Multiplying dollars by dollars would lead to square dollars, which is a mistake (and not just because dollars are actually rectangular in shape). It is “-2 × $-2”, not “$-2 × $-2″. Money are not multiplied by money. Money are mutliplied by… number of accounts, or number of repetitions, or other dimensionless numbers.