The first thing I thought about when I read this was my recent visit to the Australian outback. We were staying near Uluru, an area originally inhabited by the Anangnu. Our guides reminded us (incessantly) that the Anangu did not believe in private property. Thus they used a different numerical system: 1, 2, 3, many. So I guess for the Anangu 2 described a quality of reality, but 4 - not so much.
Last edited by jegreenwood; Jun-17-2019 at 15:18.
Those are fractions, and of course they exist, backwards from "1" to infinity this side of zero. But not the other side of zero.
The diagonal of a square is "real." The Egyptians used lengths of rope to construct buildings from a central stake, which inscribed a circle. Reality consists of many relations (ratios) which cannot be expressed numerically.
No, numbers are not abstractions if they are used to identify or count objects. I have 3 sheep, the "three-ness" is a real description.
Last edited by millionrainbows; Jun-17-2019 at 15:36.
The ancient Greeks did not think that the length of the diagonal of a square is a "real" number in the same sense as counting numbers (or even what we would call fractions). We would call this diagonal length "irrational". They called it "incommensurable" with the length of the sides. It is literally impossible to measure the length of the diagonal in terms of any kind of subdivisions of the length of the sides (and you can prove this). The diameter or radius of a circle is similarly incommensurable with the circumference of the circle. This was a sort of spooky thing first discovered by the Pythagoreans. There are still some mathematicians who regard irrational numbers with suspicion, but they are a small minority.
As for fractions, there's a reason why a many kids have trouble learning how to use the in math classes, and it's because they are actually much more sophisticated and abstract things that adults who (mostly) learned them by rote tend to take for granted. I would even argue that the kids who have difficulty with fractions somehow realize this better than many adults, or even their teachers.
Fractions, or rational numbers, do have a certain correspondence to certain real world situations (3/4 of a pie) but not so much to others (3/4 of a bicycle). Ultimately they are numbers (in a sense) that are conjured from thin air by declaring that we want every kind of division of integers (except for division by zero) to make some kind of sense. Negative numbers are similar in that they exist simply because we want every kind of subtraction of whole numbers to exist and make sense. Negative numbers are just as "real" and sensible as positive fractions are.
You can keep going. Normally you'd say that there's no numbers whose square is a negative number. That is, until you declare that such a number exists, like the "imaginary" unit i, for example, then all of a sudden you have the complex numbers, an extension of the "real" numbers that allow you to take square roots of anything. But the so-called "imaginary" numbers are no less "real" than whole numbers, or integers or rational numbers, or real numbers.
As for the counting numbers 1, 2, 3, etc., these are abstractions too. Suppose you have two piles of things, say, three apples and three bicycles. They really don't have anything in common except for the fact that there are three of them. And these three piles (let's call these piles "sets", why not) have this same "three-ness" in common with all other sets of three things. This "three-ness" in the pile of apples is some property that is not redness, or sweetness, or shape, or anything else you can say about piles of apples. Ditto for bicycles, and for every other set of three things. So what is it then? It's sort of slippery.
There are different ways to define what we mean by "three", that aren't worth getting into here. But I think if you really try to pin it down, it'll prove to be a lot more elusive than you think.
Edit: I should add that the reason why the Greeks and even some modern mathematicians view irrational numbers with suspicions is because defining them in terms of other numbers (like integers) always involves some kind of infinite process, or infinite set, the existence of which is dubious to some people. But that's an entirely different discussion. And negative numbers really do not suffer from any kind of similar challenge to their "real-ness".
Last edited by apricissimus; Jun-17-2019 at 16:10.
The Pythagorans worshipped number, and are abstractionists by nature, and are a poor choice for representing "number as reality." Still, the diagonal of a square is a "real" thing, as is Pi and circles.
That's because fractions are not quantities, they are relations. "Half" could mean 1/2 a million, 1/2 of ten, half of any quantity.As for fractions, there's a reason why a many kids have trouble learning how to use them in math classes, and it's because they are actually much more sophisticated and abstract things that adults who (mostly) learned them by rote tend to take for granted. I would even argue that the kids who have difficulty with fractions somehow realize this better than many adults, or even their teachers.
That's determined by the nature of the objects, not numbers: a bicycle is considered to be a "unity" as an object, almost like it was a "being" like a human. Fractions are best applied to QUANTITIES of things which are inherently not unified, can be "used" as materials, like pies. Why else?Fractions, or rational numbers, do have a certain correspondence to certain real world situations (3/4 of a pie) but not so much to others (3/4 of a bicycle).
Yes, but I am emphasizing how they are NOT, not how they ARE similar.Ultimately they are numbers (in a sense) that are conjured from thin air by declaring that we want every kind of division of integers (except for division by zero) to make some kind of sense. Negative numbers are similar in that they exist simply because we want every kind of subtraction of whole numbers to exist and make sense. Negative numbers are just as "real" and sensible as positive fractions are.
You're already getting lost in more abstraction. Keep it simple.You can keep going. Normally you'd say that there's no numbers whose square is a negative number. That is, until you declare that such a number exists, like the "imaginary" unit i, for example, then all of a sudden you have the complex numbers, an extension of the "real" numbers that allow you to take square roots of anything. But the so-called "imaginary" numbers are no less "real" than whole numbers, or integers or rational numbers, or real numbers.
No more slippery than any other quality. "Three-ness" is easy, and real.As for the counting numbers 1, 2, 3, etc., these are abstractions too. Suppose you have two piles of things, say, three apples and three bicycles. They really don't have anything in common except for the fact that there are three of them. And these three piles (let's call these piles "sets", why not) have this same "three-ness" in common with all other sets of three things. This "three-ness" in the pile of apples is some property that is not redness, or sweetness, or shape, or anything else you can say about piles of apples. Ditto for bicycles, and for every other set of three things. So what is it then? It's sort of slippery.
Oh, I'm sure there are, in the abstract world of numbers.There are different ways to define what we mean by "three", that aren't worth getting into here. But I think if you really try to pin it down, it'll prove to be a lot more elusive than you think.
Last edited by millionrainbows; Jun-17-2019 at 16:27.
Your ears are internally pressurized with atmospheric pressure. They do not detect absolute pressure, they detect the pressure difference between their internal pressure and the external pressure. Sound is a compression wave which causes pressure to fluctuations above and below mean atmospheric pressure so your ears detect positive and negative excursions of relative pressure. Your ear drum is effectively "sucked out" during the negative phase of a sound wave when external pressure is below the internal pressure of the ear. It is therefore totally appropriate for an acoustic signal oscillate between positive and negative values.
You can reach the same conclusion by considering how sound is produced. Sound is produced by vibration of the instrument. In a stringed instrument the sounding board moves up and down in response to the vibration of the string, producing positive and negative perturbation to the pressure. The same is true of a loudspeaker cone, with moves forward and backward producing positive and negative pressure perturbations.
It is also true that there is no need for an audio signal to be bipolar. If you were to build an audio amplifier where +10V was the equilibrium output and the signal fluctuated above and below 10V the effect would be the same. The speaker cone would move forward and backward with respect to its equilibrium position creating positive and negative pressure fluctuations. You would, however, needlessly dissipate a lot of energy in your output transistors and speaker coils.
Last edited by Baron Scarpia; Jun-20-2019 at 19:54.
Sort of like this.
parts-of-a-wave.gif
Last edited by jegreenwood; Jun-21-2019 at 14:53.
Read again. The ear is not filled with vacuum, it's internal pressure is the mean atmospheric pressure (maintained by the eustachian tubes). When the trough of the sound wave brings the external pressure below mean atmospheric pressure the ear drum is "sucked out." If you like you can say that it is pushed out by the internal pressure, which exceeds the external pressure. But that's what "suck" means. When you suck liquid through a straw there is no negative pressure, there is pressure lower than the ambient atmospheric pressure.
Last edited by Baron Scarpia; Jun-21-2019 at 16:56.
But both conditions, normal or higher pressure, are both "positive" values. There is no negative pressure, as you say. The sound wave is acting upon values of pressure which are all positive.
Mean atmospheric pressure is a force which is constantly positive, and which is "pushing in" on the eardrum. The natural state of the eardrum, then would be no pressure at all. So all of this "pushing and pulling" and "waves and troughs" are occurring under atmospheric pressure which is, to begin with, a state of positive pressure. Anything less than that is not a "negative" value, only a lesser positive one.