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What is "real"? The ancients may have understood negative numbers, but IIRC the concept of zero (additive identity) wasn't understood until Arabic Numbers / Al

What is "real"? The ancients may have understood negative numbers, but IIRC the concept of zero (additive identity) wasn't understood until Arabic Numbers / Algebra. So "zero" isn't real, at least not to the Romans or Greeks. Once Algebra is invented, we learn that not only is "zero" a useful concept (additive identity), but so is "one" (multiplicative identity). More importantly, the inverse of multiplication can be defined as "blah times number == 1". IIRC, the Greeks somehow understood this concept before the invention of Algebra, but its easier to see once you have Algebra. Once we have those, its not too difficult to extend into irrational numbers, such as sqrt(2). The Greeks almost discovered it, but were too stubborn. There's a lot of discussion in Greek Mathematics over sqrt(2)... they're so so close but they were too much of a fanboy of integers / rational numbers to actually reach the concept. Once we understand sqrt(2), its not too difficult to extend the principle to sqrt(-1) or sqrt(-2), reaching the concept of "i". ----------- sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering (https://en.wikipedia.org/wiki/Phasor https://en.wikipedia.org/wiki/Phasor). Its as "real" as any other number. IE: We can truly measure real-world objects (ie: the voltage going through your power-lines) and assign them a value of "i", and use that to predict what happens to the electricity as it flows through your equipment and/or house. So all of these things are "real" numbers, that describe truly physical real-world objects. Its just an unfortunate naming convention of "Real" vs "Imaginary" vs "Rational" vs "Irrational", which comes down to history / order of discoveries, rather than usefulness.

chriswarbo 4y ago
> sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering I'd say sqrt(-1) describes "real" voltages / currents; but we can find other models which don't involve sqrt(-1). Those models may be more complicated, and have fewer rich connections to pre-existing work, but they're just as valid. This is usually where appeals to Occam's Razor come in, and physicists start appealing to "elegance", "beauty", etc.
- dragontamer 4y ago
  You're not going to be able to describe how voltages/currents work in an AC circuit unless you reach into 2-vectors / matrices. In Complex Arithmetic, we call it the "imaginary axis". But reach into Linear Algebra and you'd just call it a "2nd dimension". When things oscillate (and AC current is oscillating), there's just certain mathematical properties that occur. It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly. Or, have a 2-vector linear-algebra representation, since its equivalent. EDIT: I guess there's the differential-equations representation, but I feel that reaches into linear-algebra very, very quickly... or complex numbers. (E^(pi * i) keeps coming up...)
  - DontchaKnowit 4y ago
    I have never seen complex numbers represented via linear algebra... would love a crash course on this as I find reasoning about complex numbers to be very difficult and find linear algebra to be imminently logical in all cases.
    - dragontamer 4y ago
      A complex number is "just" a 2-vector. Instead of calling the two dimensions "x and y", consider them "(unnamed) and i". For example: the vector <1, 1> is equivalent to 1 + i. The complex conjugate is a reflection across the x axis: <1,1> into <1, -1>, or in the Complex representation from 1+i into 1-i. Addition of numbers progresses just like addition of vectors: <1,1> + <2,2> == <3,3>. Similarly: (1+i) + (2+2i) == (3+3i). -------- IIRC, Multiplication of Complex numbers is a rotation across the two axis. The overall concept is pretty difficult to describe succinctly, but it is "obvious" in a simple case. Multiplication by "i" is just a 90-degree rotation counter-clockwise. That is: <1, 1> rotated 90-degrees is <-1, 1>. Or 1+i rotated 90-degrees is (i * (1+i)) == (-1 + i). Rotating again is another multiplication of i, or i * (-1 + i) == (-1 - i) == <-1, -1> in vector form. Then <-1, 1> and finally <1, 1> again (4x 90-degree rotations == 360 degrees). -------- Etc. etc. etc. Everything you do in complex arithmetic has a good interpretation in the vector / geometric space. A lot of the intuition needs to be built from the polar form of complex numbers: r * e^(theta * i), since the use of complex numbers is largely about rotations about the axis.
lupire 4y ago
The difference is that the probability of any given "real" number being used for anything, ever, or even being nameable, or even being potential covered by a naming scheme, is 0. This is also true for countably infinite collections of real numbers. Real numbers are unreal, at a scale incomprehensibly beyond the merely slightly hidden numbers like complex algebraics and the known transcendentals. Real numbers aren't numbers. R is a background continuum in which numbers exist. It is almost entirely void, yet contains an infinite collection of actual numbers.
- dragontamer 4y ago
  Pi has a name. As does sqrt(2). As does "e". All of the important numbers have a name. In fact, we can do funny things with these names, such as (e^(pi * i)) == -1, and they're fully valid. "Lack of names" seems like a non-issue. Plenty of irrational numbers have names. Heck, plenty of imaginary numbers have names (ex: i). ------ We have plenty of numbers we can't even write down. Such as Graham's number. Or googleplex. Does that make these numbers "less real" ??
  - thethirdone 4y ago
    You in fact did write down "Graham's number" and "googleplex". Most real numbers cannot even be written down as a finite sequence of symbols in any way. However for algebraic numbers ("Plenty of irrational numbers"), every single one has a finite description. There are uncountably many reals and only a countable number of "names" we can give them. This is a very real problem that makes people say that the real numbers are not real.
    - dragontamer 4y ago
      > finite description Oh jeez, its like the space of rational numbers is well described by Arabic Numerals or something. And that anything that's a non-rational number cannot be described by Arabic Numerals + decimal points. :-p Don't confuse our current way of writing down numbers with alternative writing methods. That's my point. "Rational numbers" are simply one-and-the-same as "The space described by finite-length Arabic Numerals + decimal points", no more, no less. EDIT: To put it another way: you're describing a limitation of Arabic Numerals. Not a limitation to math in general. Much like how Roman Numerals are a terrible representation of modern math, we need other representations (ie: Algebraic notation) to represent other concepts in modern math.
pmoriarty 4y ago
The map is not the territory.
constantcrying 4y ago
>The ancients may have understood negative numbers, but IIRC the concept of zero (additive identity) wasn't understood until Arabic Numbers / Algebra. So "zero" isn't real, at least not to the Romans or Greeks. The idea of the point as a pseudo-algebraic value was very much prominent in euclids elements. The symbol came later, but of course the concept was very much established. It is preposterous to attribute to a civilization which, in Archimedes, discovered calculus, that they did not understand the concept of zero. They just thought far more geometrically than most people are used to. Also by "real" he means cauchy sequences of rationals, or some equivalent definition.

chriswarbo 4y ago

> sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering I'd say sqrt(-1) describes "real" voltages / currents; but we can find other models which don't involve sqrt(-1). Those models may be more complicated, and have fewer rich connections to pre-existing work, but they're just as valid. This is usually where appeals to Occam's Razor come in, and physicists start appealing to "elegance", "beauty", etc.

dragontamer 4y ago
You're not going to be able to describe how voltages/currents work in an AC circuit unless you reach into 2-vectors / matrices. In Complex Arithmetic, we call it the "imaginary axis". But reach into Linear Algebra and you'd just call it a "2nd dimension". When things oscillate (and AC current is oscillating), there's just certain mathematical properties that occur. It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly. Or, have a 2-vector linear-algebra representation, since its equivalent. EDIT: I guess there's the differential-equations representation, but I feel that reaches into linear-algebra very, very quickly... or complex numbers. (E^(pi * i) keeps coming up...)
- DontchaKnowit 4y ago
  I have never seen complex numbers represented via linear algebra... would love a crash course on this as I find reasoning about complex numbers to be very difficult and find linear algebra to be imminently logical in all cases.
  - dragontamer 4y ago
    A complex number is "just" a 2-vector. Instead of calling the two dimensions "x and y", consider them "(unnamed) and i". For example: the vector <1, 1> is equivalent to 1 + i. The complex conjugate is a reflection across the x axis: <1,1> into <1, -1>, or in the Complex representation from 1+i into 1-i. Addition of numbers progresses just like addition of vectors: <1,1> + <2,2> == <3,3>. Similarly: (1+i) + (2+2i) == (3+3i). -------- IIRC, Multiplication of Complex numbers is a rotation across the two axis. The overall concept is pretty difficult to describe succinctly, but it is "obvious" in a simple case. Multiplication by "i" is just a 90-degree rotation counter-clockwise. That is: <1, 1> rotated 90-degrees is <-1, 1>. Or 1+i rotated 90-degrees is (i * (1+i)) == (-1 + i). Rotating again is another multiplication of i, or i * (-1 + i) == (-1 - i) == <-1, -1> in vector form. Then <-1, 1> and finally <1, 1> again (4x 90-degree rotations == 360 degrees). -------- Etc. etc. etc. Everything you do in complex arithmetic has a good interpretation in the vector / geometric space. A lot of the intuition needs to be built from the polar form of complex numbers: r * e^(theta * i), since the use of complex numbers is largely about rotations about the axis.

lupire 4y ago

The difference is that the probability of any given "real" number being used for anything, ever, or even being nameable, or even being potential covered by a naming scheme, is 0. This is also true for countably infinite collections of real numbers. Real numbers are unreal, at a scale incomprehensibly beyond the merely slightly hidden numbers like complex algebraics and the known transcendentals. Real numbers aren't numbers. R is a background continuum in which numbers exist. It is almost entirely void, yet contains an infinite collection of actual numbers.

dragontamer 4y ago
Pi has a name. As does sqrt(2). As does "e". All of the important numbers have a name. In fact, we can do funny things with these names, such as (e^(pi * i)) == -1, and they're fully valid. "Lack of names" seems like a non-issue. Plenty of irrational numbers have names. Heck, plenty of imaginary numbers have names (ex: i). ------ We have plenty of numbers we can't even write down. Such as Graham's number. Or googleplex. Does that make these numbers "less real" ??
- thethirdone 4y ago
  You in fact did write down "Graham's number" and "googleplex". Most real numbers cannot even be written down as a finite sequence of symbols in any way. However for algebraic numbers ("Plenty of irrational numbers"), every single one has a finite description. There are uncountably many reals and only a countable number of "names" we can give them. This is a very real problem that makes people say that the real numbers are not real.
  - dragontamer 4y ago
    > finite description Oh jeez, its like the space of rational numbers is well described by Arabic Numerals or something. And that anything that's a non-rational number cannot be described by Arabic Numerals + decimal points. :-p Don't confuse our current way of writing down numbers with alternative writing methods. That's my point. "Rational numbers" are simply one-and-the-same as "The space described by finite-length Arabic Numerals + decimal points", no more, no less. EDIT: To put it another way: you're describing a limitation of Arabic Numerals. Not a limitation to math in general. Much like how Roman Numerals are a terrible representation of modern math, we need other representations (ie: Algebraic notation) to represent other concepts in modern math.

pmoriarty 4y ago

The map is not the territory.

constantcrying 4y ago

>The ancients may have understood negative numbers, but IIRC the concept of zero (additive identity) wasn't understood until Arabic Numbers / Algebra. So "zero" isn't real, at least not to the Romans or Greeks. The idea of the point as a pseudo-algebraic value was very much prominent in euclids elements. The symbol came later, but of course the concept was very much established. It is preposterous to attribute to a civilization which, in Archimedes, discovered calculus, that they did not understand the concept of zero. They just thought far more geometrically than most people are used to. Also by "real" he means cauchy sequences of rationals, or some equivalent definition.