It's also a complex number, a Unicode character, an ASCII character, an Extended ASCII character, a glyph, the multiplicative identity element, a raster image, ...
The GP point is correct; we implicitly convert between all these representations naturally and quickly, but there are interesting branches of mathematics that consider those conversions explicitly and find nuances (eg, category theory).
All the examples of shadows mentioned by Douglas in the video [0]:
Traditional Shadow from Sunlight: A tree casting a shadow in bright sunlight during the summer, characterized by the typical darkening on the ground.
Snow Shadow: Occurs in winter when a tree intercepts snowflakes, resulting in an absence of snow under the tree despite being surrounded by snow. This is likened to a shadow because it mimics the visual absence typical of shadows, even though caused by a different kind of blockage.
Rain Shadow: Explained with the example of the Cascade Mountains, where the mountains block rain clouds leading to dry conditions on the leeward side, effectively creating a "shadow" where rain is absent due to the geographical barrier.
Car Shadow by a Truck: Described during highway driving, where a large truck casts a "shadow" by preventing cars behind it from passing. This "shadow" is the area in front of the truck where cars tend to stay clear due to the difficulty in passing the truck.
Shadow Cast by England Due to the Gulf Stream: This is a metaphorical shadow, where England blocks the warm Gulf Stream from reaching certain areas, resulting in colder temperatures on one side, while allowing it to pass above to northern Norway, warming it. This is referred to as the "shadow of England" on the coast of Norway, influenced by the flow of water rather than light.
These examples use the concept of "shadow" in various physical and metaphorical contexts, showing disruptions in different types of flows (light, snow, rain, traffic, and ocean currents) caused by obstructions.
For example, the "2" in "2π" is not the same type of "2" as in x^2 or 2x generally. Yet, physicists (to pick a random group) will blend in these factors, resulting in nonsense. As a random example, one of the Einstein field equations has "8π" in it. Eight what!? What aspect of the universe is this counting out eight of -- a weirdly large integer constant? This actually ought to be "4(2pi)", and then "4" is the number of spacetime dimensions, which makes a lot more sense.
Similarly, in at least one place the square of the pseudoscalar (I^2) was treated as a plain -1 integer constant and accidentally "folded" into other unrelated integer constants. This causes issues when moving from 2D to 3D to 4D.
These examples miss the mark somewhat. The "2" in "2π" can mean several things (the nonnegative integer 2, the integer 2, the rational 2, the real 2) that are all commonly identified but are different. The "2" in "x^2" usually means the nonnegative integer 2. The "2" in "2x" can usually mean the nonnegative integer or the integer 2, but also the other 2's depending on what x is. The problem is not that the meaning of 2 varies across different expressions, but that it can vary within each single expression.
The best example is perhaps the polynomial ring R[x][y], which consists of polynomials in the variable y over the ring of polynomials in the variable x over the real numbers. Any algebraist would tell you that it is obviously just the two-variable polynomial ring R[x, y] in disguise, because you can factor out all the y-powers and then the coefficients will be polynomials in x. But the rings are very much not the same at the level of implementation, and every time you use their "equality" (canonical isomorphy), you need to keep the actual conversion map (the isomorphism) in the back of your mind.
Haskell shows (one way of) how you can have numerical literals like 2 that can be used with many different types, but still be strongly statically typed.
That by itself isn't a problem. But making all the other confusions you mention is a problem.
Saying 1 is both an integer and a rational number is wildly different from saying it is both an integer and an ASCII character. Z is a subset of Q. ASCII characters don’t overlap with either.
When you construct numbers using sets under ZFC axioms or inside lambda calculus what you get is representation. But 1 is just 1.
Your keyboard has a button with ‘1’ printed on it. When you push that, you don’t always get an integer or a rational number. You can convert what you get to either. So there must be overlap with ASCII somehow?
But the integers are a subset of the rationals, which are a subset of the reals, which are a subset of the complex numbers. Looking only at the objects and not their operations 1 (integer) = 1 (rational) = 1 (real) = 1 (complex). Moreover, when we do account for the operations, we also see that 1 + 1 = 2 and 1 * 1 = 1 in every one of those systems. This isn't just a coincidence, of course; it's by design.
However, the way you arrive at 1 + 1 = 2 is not the same (though I suppose you could short-circuit the algorithm). Rational addition requires finding a common denominator, while integer addition doesn't. They achieve the same result when the inputs are integers, and again this is by design, but the process isn't the same. Ditto real addition vs. rational and complex addition vs. real.
In higher-level mathematics, the operations on the objects become definitional. We don't look at just a set of things, we look at a set of things and the set of operations upon those things. Thus "1 with integer addition and integer multiplication" becomes the object under consideration (even if it's just contextually understood) instead of simply 1. This is why they don't satisfy higher-level notions of equivalence, even if they intentionally do satisfy simple equality as taught in grade school.
Of course, the entire point of the submitted paper is to examine this in detail.
> But the integers are a subset of the rationals, which are a subset of the reals, which are a subset of the complex numbers.
It depends on definitions, and, in some sense, the point of the common approach to mathematics is not just that one does not, but that one cannot, ask such questions. One approach is to look at natural numbers set theoretically, starting with 0 = ∅; to define integers as equivalence classes of pairs of natural numbers; to define rational numbers as equivalence classes of certain pairs of integers; and to define real numbers as equivalence classes of Cauchy sequences of rational numbers. In each of these cases there is an obvious injection which we are used to regarding as inclusion, but most of mathematics is set up to make it meaningless even to ask whether the natural number 1 is the same as the integer 1 is the same as ….
That is to say, if you're working on an application where encoding details are important, then you can and will ask such questions; but if I am writing a paper about natural numbers, I do not have to worry about the fact that, for some choice of encoding, the number 2 = {∅, {∅}} is the same as the ordered pair (0, 0) = {0, {0, 0}} = {∅, {∅}}, and in fact it is meaningless to test whether 2 "equals" (0, 0). The philosophy of studiously avoiding such meaningless questions leads some to avoid even testing for equality, as opposed to isomorphism; failing to do so used to be referred to in category-theoretic circles as "evil", although, as the nLab points out if you try to go to https://ncatlab.org/nlab/show/evil , it seems common nowadays to avoid such language.
This is not the point of the article. Even at the level of the objects themselves, 1 : integer is not 1 : rational. The latter is an ordered pair (1, 1) of two coprime positive integers, or an equivalence class of ordered pairs up to cancelling. Some ugly hackery is required to really make the integers equal to their respective rationals, and its consequences aren't great either (just imagine that some rationals are pairs while others are not -- that's what you get if you forcibly replace the rational k/1 by the integer k), and no one wants to do that.
There are multiple things that we denote using a particular symbol: 1. 1 in and of itself is not a single concept. You could replace the symbol 1 with any other, even the sound of a fart and the concepts still remain the same. Given that we somehow agree that a fart sound shall be the way that we refer to a concept/object/thing.
It's a largely useful conceit to use 1 for all of those objects and more besides. It makes talking about them easier but we do have to be careful to use the correct rule-set for their manipulation.
I would personally prefer to see 1.0 for "rational 1" or perhaps 1. but that would require a convoluted sentence to avoid 1. being at the end of the sentence, unless we allow for: 1.! Well, that would work but what about 1.? Oh for ffs, I mean: 1..
One notes that one's own ones may not be the same as one's other ones.
If only I could spin "won", "own" and perhaps "wan" into that last sentence! ... Right, I've wedged in own. Needs some work 8)
Well, you give give 1.0 for the rational one. But what about the real or complex one? Or what about the Gaussian Integer? What about the 2-adic one, or the 3-adic one, or any other p-adic one? What about the different kinds of floating point numbers?
I don't think the rational one is special enough that we need to different notation just for her and for her alone. (Though that specific distinction can make sense in some contexts. Just not universally.)
It's also a complex number, a Unicode character, an ASCII character, an Extended ASCII character, a glyph, the multiplicative identity element, a raster image, ...
The GP point is correct; we implicitly convert between all these representations naturally and quickly, but there are interesting branches of mathematics that consider those conversions explicitly and find nuances (eg, category theory).
All the examples of shadows mentioned by Douglas in the video [0]:
Traditional Shadow from Sunlight: A tree casting a shadow in bright sunlight during the summer, characterized by the typical darkening on the ground.
Snow Shadow: Occurs in winter when a tree intercepts snowflakes, resulting in an absence of snow under the tree despite being surrounded by snow. This is likened to a shadow because it mimics the visual absence typical of shadows, even though caused by a different kind of blockage.
Rain Shadow: Explained with the example of the Cascade Mountains, where the mountains block rain clouds leading to dry conditions on the leeward side, effectively creating a "shadow" where rain is absent due to the geographical barrier.
Car Shadow by a Truck: Described during highway driving, where a large truck casts a "shadow" by preventing cars behind it from passing. This "shadow" is the area in front of the truck where cars tend to stay clear due to the difficulty in passing the truck.
Shadow Cast by England Due to the Gulf Stream: This is a metaphorical shadow, where England blocks the warm Gulf Stream from reaching certain areas, resulting in colder temperatures on one side, while allowing it to pass above to northern Norway, warming it. This is referred to as the "shadow of England" on the coast of Norway, influenced by the flow of water rather than light.
These examples use the concept of "shadow" in various physical and metaphorical contexts, showing disruptions in different types of flows (light, snow, rain, traffic, and ocean currents) caused by obstructions.
[0] : https://youtu.be/LnTVzB5S8m4?t=973
Mathematics is surprisingly weakly typed.
For example, the "2" in "2π" is not the same type of "2" as in x^2 or 2x generally. Yet, physicists (to pick a random group) will blend in these factors, resulting in nonsense. As a random example, one of the Einstein field equations has "8π" in it. Eight what!? What aspect of the universe is this counting out eight of -- a weirdly large integer constant? This actually ought to be "4(2pi)", and then "4" is the number of spacetime dimensions, which makes a lot more sense.
Similarly, in at least one place the square of the pseudoscalar (I^2) was treated as a plain -1 integer constant and accidentally "folded" into other unrelated integer constants. This causes issues when moving from 2D to 3D to 4D.
These examples miss the mark somewhat. The "2" in "2π" can mean several things (the nonnegative integer 2, the integer 2, the rational 2, the real 2) that are all commonly identified but are different. The "2" in "x^2" usually means the nonnegative integer 2. The "2" in "2x" can usually mean the nonnegative integer or the integer 2, but also the other 2's depending on what x is. The problem is not that the meaning of 2 varies across different expressions, but that it can vary within each single expression.
The best example is perhaps the polynomial ring R[x][y], which consists of polynomials in the variable y over the ring of polynomials in the variable x over the real numbers. Any algebraist would tell you that it is obviously just the two-variable polynomial ring R[x, y] in disguise, because you can factor out all the y-powers and then the coefficients will be polynomials in x. But the rings are very much not the same at the level of implementation, and every time you use their "equality" (canonical isomorphy), you need to keep the actual conversion map (the isomorphism) in the back of your mind.
Haskell shows (one way of) how you can have numerical literals like 2 that can be used with many different types, but still be strongly statically typed.
That by itself isn't a problem. But making all the other confusions you mention is a problem.
Saying 1 is both an integer and a rational number is wildly different from saying it is both an integer and an ASCII character. Z is a subset of Q. ASCII characters don’t overlap with either.
When you construct numbers using sets under ZFC axioms or inside lambda calculus what you get is representation. But 1 is just 1.
Your keyboard has a button with ‘1’ printed on it. When you push that, you don’t always get an integer or a rational number. You can convert what you get to either. So there must be overlap with ASCII somehow?
7 replies →
But the integers are a subset of the rationals, which are a subset of the reals, which are a subset of the complex numbers. Looking only at the objects and not their operations 1 (integer) = 1 (rational) = 1 (real) = 1 (complex). Moreover, when we do account for the operations, we also see that 1 + 1 = 2 and 1 * 1 = 1 in every one of those systems. This isn't just a coincidence, of course; it's by design.
However, the way you arrive at 1 + 1 = 2 is not the same (though I suppose you could short-circuit the algorithm). Rational addition requires finding a common denominator, while integer addition doesn't. They achieve the same result when the inputs are integers, and again this is by design, but the process isn't the same. Ditto real addition vs. rational and complex addition vs. real.
In higher-level mathematics, the operations on the objects become definitional. We don't look at just a set of things, we look at a set of things and the set of operations upon those things. Thus "1 with integer addition and integer multiplication" becomes the object under consideration (even if it's just contextually understood) instead of simply 1. This is why they don't satisfy higher-level notions of equivalence, even if they intentionally do satisfy simple equality as taught in grade school.
Of course, the entire point of the submitted paper is to examine this in detail.
> But the integers are a subset of the rationals, which are a subset of the reals, which are a subset of the complex numbers.
It depends on definitions, and, in some sense, the point of the common approach to mathematics is not just that one does not, but that one cannot, ask such questions. One approach is to look at natural numbers set theoretically, starting with 0 = ∅; to define integers as equivalence classes of pairs of natural numbers; to define rational numbers as equivalence classes of certain pairs of integers; and to define real numbers as equivalence classes of Cauchy sequences of rational numbers. In each of these cases there is an obvious injection which we are used to regarding as inclusion, but most of mathematics is set up to make it meaningless even to ask whether the natural number 1 is the same as the integer 1 is the same as ….
That is to say, if you're working on an application where encoding details are important, then you can and will ask such questions; but if I am writing a paper about natural numbers, I do not have to worry about the fact that, for some choice of encoding, the number 2 = {∅, {∅}} is the same as the ordered pair (0, 0) = {0, {0, 0}} = {∅, {∅}}, and in fact it is meaningless to test whether 2 "equals" (0, 0). The philosophy of studiously avoiding such meaningless questions leads some to avoid even testing for equality, as opposed to isomorphism; failing to do so used to be referred to in category-theoretic circles as "evil", although, as the nLab points out if you try to go to https://ncatlab.org/nlab/show/evil , it seems common nowadays to avoid such language.
This is not the point of the article. Even at the level of the objects themselves, 1 : integer is not 1 : rational. The latter is an ordered pair (1, 1) of two coprime positive integers, or an equivalence class of ordered pairs up to cancelling. Some ugly hackery is required to really make the integers equal to their respective rationals, and its consequences aren't great either (just imagine that some rationals are pairs while others are not -- that's what you get if you forcibly replace the rational k/1 by the integer k), and no one wants to do that.
There are multiple things that we denote using a particular symbol: 1. 1 in and of itself is not a single concept. You could replace the symbol 1 with any other, even the sound of a fart and the concepts still remain the same. Given that we somehow agree that a fart sound shall be the way that we refer to a concept/object/thing.
It's a largely useful conceit to use 1 for all of those objects and more besides. It makes talking about them easier but we do have to be careful to use the correct rule-set for their manipulation.
I would personally prefer to see 1.0 for "rational 1" or perhaps 1. but that would require a convoluted sentence to avoid 1. being at the end of the sentence, unless we allow for: 1.! Well, that would work but what about 1.? Oh for ffs, I mean: 1..
One notes that one's own ones may not be the same as one's other ones.
If only I could spin "won", "own" and perhaps "wan" into that last sentence! ... Right, I've wedged in own. Needs some work 8)
Well, you give give 1.0 for the rational one. But what about the real or complex one? Or what about the Gaussian Integer? What about the 2-adic one, or the 3-adic one, or any other p-adic one? What about the different kinds of floating point numbers?
I don't think the rational one is special enough that we need to different notation just for her and for her alone. (Though that specific distinction can make sense in some contexts. Just not universally.)