Take the ubiquitous example of a count of apples. It's obvious and natural to us what it means for me to have a positive number of apples, it's something we can count. — Jerry
"But," you might say, "none of this shakes my belief that 2 and 2 are 4." You are quite right, except in marginal cases -- and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a meter. Two must be two of something, and the proposition "2 and 2 are 4" is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. "Well, at any rate there are four animals," you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. "Well, then living organisms," you say. But there are things of which it is doubtful whether they are living organisms or not. You will be driven into saying: "Two entities and two entities are four entities." When you have told me what you mean by "entity," we will resume the argument. — Russell
(Remarks on the Foundations of Mathematics [RFM] I-113).However many rules you give me—I give a rule which justifies my employment of your rules — Wittgenstein
Take the ubiquitous example of a count of apples. It's obvious and natural to us what it means for me to have a positive number of apples, it's something we can count. It's less obvious what it might mean for there to be an amount which is less than nothing. — Jerry
If you take out $100 dollars from your account of +$50, you may end up with -$50 dollars, but this wasn't a ×(-1) operation, that was a subtract $100 operation, which happens to yield the same result. — Jerry
My first argument is that our intuitions of what negative values mean, and especially the operations between them, are sloppy and imprecise. Take the ubiquitous example of a count of apples. It's obvious and natural to us what it means for me to have a positive number of apples, it's something we can count. It's less obvious what it might mean for there to be an amount which is less than nothing. The first instinct most people have to apply negative numbers to such a situation is to introduce debt. Debt is a fine use-case of negative arithmetic, but offering it as a tangible realization of negative numbers in a realist sense, the sense in which we count positive numbers of apples, is rather insidious. — Jerry
To demonstrate my point, consider this: in the usual examples of negative numbers in nature (temperature, debt, sea level) when does the ×(-1) operation occur? For example, sea level may go up and down (add/subtract) in increments, but does the sea level ever flip from above to below? If you take out $100 dollars from your account of +$50, you may end up with -$50 dollars, but this wasn't a ×(-1) operation, that was a subtract $100 operation, which happens to yield the same result. — Jerry
Think of 2 representing the height of a mound of dirt and -2 representing the depth of a hole beside it. — Pie
Now it could be the case that regular counting has its own context, which I feel is eluded to indirectly by , although I don't yet understand the meaning of the quotes they provided. To quickly reiterate, it's not that I think negative numbers can't refer to things in nature, it just seems like extra steps are needed to make them make sense, which makes them somewhat different from positive numbers.The main difference I perceive is that negative numbers require a context within which to function, unlike positive amounts which I seem to be able to measure or count in any situation. . .I think that this required context does make negative numbers at least seem one step removed from the naturalness of the positives. — Jerry
As for multiplying by a negative, it's not hard to find examples. — Banno
:up:Friend, there are many interesting questions and debates involved with the foundations of math . . . The existence of negative numbers is not one of them. — Real Gone Cat
Anadi has no beginning, but has an end () — Wikipedia
it's not that I think negative numbers can't refer to things in nature, it just seems like extra steps are needed to make them make sense, which makes them somewhat different from positive numbers. — Jerry
The point is that we are using sloppy intuition to justify the rules of negatives, intuition that clearly didn't convince mathematicians of the past, and perhaps there's some value in recognizing that. — Jerry
Negative numbers, as some members have already realized, are simply extensions of numerical patterns, not forwards like how we're so habituated to doing but backwards. — Agent Smith
Good point! In Physics, changes in value can only proceed "forward" (positive) one-step-at-a-time. But in meta-physical*1 Mathematics, we can imagine the number-line as a whole, and see both forward (future) and backward (past) at a glance. Likewise, we can imagine Time as a number-line, allowing us to follow it back to the beginning of time . . . and beyond. That's why Physicists can only work on the here & now, while Cosmologists & Sci-Fi-ers can speculate on Multiverses-without-beginning and Many-Worlds-without-location. Such conjectures are mathematical concepts instead of physical observations. :smile:Negative numbers, as some members have already realized, are simply extensions of numerical patterns, not forwards like how we're so habituated to doing but backwards. — Agent Smith
Most interesting. — Ms. Marple
Flipping (reflecting) alias rotating (turning) by π radians is a good geometric way to grasp what negative numbers are. — Agent Smith
The words "positive" and "negative" have connotations good (ethical) and bad (unethical). Do these auxiliary meanings have mathematical origins or is there some other nonmathematical explanation as to why? — Agent Smith
When I say "Eat!" I am encouraging you to eat (positive)
But when I say "Do not eat!" I am saying the opposite (negative).
Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).
The tank has 30,000 liters, and 1,000 liters are taken out every day. What was the amount of water in the tank 3 days ago?
We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go back 3 days), so the change is:
−3 × −1,000 = +3,000
The full calculation is:
30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000
So 3 days ago there were 33,000 liters of water in the tank.
We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go back 3 days), so the change is:
−3 × −1,000 = +3,000
Sorry, I'm done. A basic math course might help.
Then again, it might not. — Real Gone Cat
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