This kind of sloppy writing is too frequently encountered and it is annoying: "Archimedes didn’t think of 22/7, 223/71, or pi as numbers; to him, they were ratios of magnitudes".
This has nothing to do with "thinking". This is just about (incorrect) language translation and mathematical terminology.
In Ancient Greek and Latin and in any other old languages, the word "number" designated the result of a counting operation, i.e. it corresponds to what in modern mathematical terminology is named "natural number", so it must be translated as such in a mathematical context.
In Ancient Greek and Latin, the word "magnitude" (or "measure") designated the result of a measurement operation, like the measurement of a length with a standard ruler or the measurement of a weight with weighing scales, i.e. it corresponds to what in modern mathematical terminology is named "real number", so it must be translated as such in a mathematical context.
There is no difference between Archimedes and a modern mathematician, both distinguish natural numbers and real numbers, but since they speak different languages, they use different words for these 2 concepts. Whenever you read an ancient mathematical text, "number" must be understood as "natural number", while "magnitude" or "measure" must be understood as "real number".
The names do not matter, this is just a problem of language translation (which is done usually incorrectly for texts with mathematical terms, like also for texts with specialized terms from physics, chemistry, biology or mineralogy, because the translators have little knowledge about those domains).
The important difference between ancient mathematics and modern mathematics is that the ancients did not have a method of construction of the real numbers from natural numbers, which has been conceived only in the 19th century, by the time of Cantor, Dedekind etc. The ancients also did not have the method of abstract algebra, where you prove something for a set of axioms of a given form (e.g. of group, of ring, of field etc.) and then the proof is valid for any set where that set of axioms is satisfied. Abstract algebra is also a creation of the 19th century.
Because of that, the ancients had 2 independent sets of axioms for natural numbers and for real numbers, even if many of them were identical in form. The consequence of this was that many theorems and demonstrations had to be duplicated for natural numbers and for real numbers, because something proven for one of them could not be applied directly to the other.
I think there's a sense in which moderns feel that the reals (or at least the rationals) are a natural category: that 5 and 0.3 are the same "kind of thing". Mathematicians talk about the distinction between different classes of numbers, but to most people they're all numbers.
Do you think that sense was shared by the ancients, or do you think that the linguistic distinction mirrored a stronger mental separation between the two? It sounds like it might have done if they had to do the work of duplicating proofs from one to the other. Did they have a single word to describe the shared concept?
To the ancients, as already clearly expressed by Aristotle, natural numbers and real numbers were both quantities. Quantities were classified in discrete quantities, of which natural numbers are an example and continuous quantities, of which real numbers are an example. This classification remains completely valid today.
The ancient mathematicians and philosophers were using the word "quantity" in most contexts where modern people use the word "number", i.e. when the word is applied to different kinds of "numbers", not just to natural numbers.
There is no difference in thinking between ancients and moderns, it is just a difference in the words that happen to be used.
Both the similarities and the differences between natural numbers and real numbers are well entrenched in most natural human languages since many millennia ago, before any scientific theory of quantities, numbers and magnitudes, as exemplified by the similarities and differences between questions like "How many ... do you have?" and "How much ... do you have?".
Actually I consider the ancient usage of the words as more sound than the modern usage. There appears little justification for the modern usage of the word "number" instead of the previous usage of "quantity", except that "number" is a shorter word than "quantity", so the change in terminology is just due to laziness, not to any theoretical reason. However what has been gained by saying "number" instead of "quantity" when the wider sense is intended, has been lost due to the requirement for qualifying "number" as "natural", "real", "integer" etc., when the narrower meaning is intended.
Etymologically, "number" is the result of counting, which real numbers and many other kinds of "numbers" that correspond to continuous quantities are not.
Suppose we designate a unit line segment, and from that we pick out a line segment of (what we now call) length 2 and separately a rectangle of area 3. To "an ancient" (like Aristotle), does it make sense to add the length and the area to get 5? If you then drew a line segment of length 5, would they say it has the same magnitude?
I've read Tau Manifesto [0], and I am now convinced that 3.14... is not the best circle constant, "turn" (tau, τ) with value 6.28... would be a much better choice.
How many radians in full circle? 1 tau (turn). What about 1/3 circle (120 degree)? 1/3 tau. What about euler's constant? e^i*tau=1, or in other words rotate vector by 1 turn and end up at start position.
So beautiful. So unreachable - pi has so much legacy, there is zero chance of changing it.
There are some advantages also for taking Pi/2 as the circle constant.
The choice between 2xPi and Pi/2 is equivalent with the choice between the cycle and the right angle as the unit of plane angle (the right angle corresponds with the "imaginary" unit, i.e. any point on the unit circle corresponds to i^x, with x in right angles). When extended to 3 dimensions, the corresponding constants for the solid angle become 4xPi and Pi/2, so Pi/2 is more consistent (it corresponds to taking the orthogonal trihedron as the unit of solid angle). The consistency of Pi/2 remains true for higher-dimensional spaces, but that has little practical importance.
"Pi" is by far the worst choice, in all computational applications either 2xPi or Pi/2 are needed, not Pi.
The only case where Pi appears naturally is in measurements, because both the circumference and the diameter are much easier to measure than the radius, and Pi is the relationship between these 2 practical measurements, allowing the conversion of one to the other.
Nowadays, the computational uses of Pi are many orders of magnitude more frequent than the conversions between the practical measurements of diameter and circumference, so the use of Pi is undesirable inside computer programs or in symbolic processing of mathematical formulae.
In general, it is much better to implement trigonometric functions where the argument is either x*Pi/2 or x*2*Pi, instead of traditional trigonometric functions, because the argument reductions are fast and exact. Sadly, the floating-point arithmetic standard defines useless functions of Pi*x, like sinPi, cosPi etc., instead of using any of the 2 better choices.
The only possible advantage of traditional trigonometric functions is at integration or differentiation, where they could save a multiplication, but in almost all applications of this kind the argument of the function is not x, but an expression with at least one multiplication, and the additional multiplication with a constant caused by using x*Pi/2 or x*2*Pi can frequently be done at compile time, or at run time, but only once, not at every computation.
Look at the usual equation: A = πr². Why is there no "2" there?
Let's derive it, and in particular, let's derive it from the onion proof, which is that a circle's area is composed of many small circles, arranged concentrically, like a 2D onion:
A = ∫_0^r 2πt dt
There's that blasted 2 again. The tau form is more beautiful:
A = ∫_0^r τt dt
Integrate it, and you'll get A = τr²/2, the constant being a result of the integral.
That is, to me, the usual equation is more properly A = 2πr²/2, the two 2s being different in their origins, and we just usually use & memorize the simplified form.
I'm still surprised that Pi isn't considered a universal physical constant, given Buffon's needle depends on (IIUC) the physical curvature of nonrelativistic space.
Is it because pi isn’t measured, but calculated? The wikipedia article (https://en.m.wikipedia.org/wiki/Physical_constant) makes a distinction between a mathematical constant and a physical constant, stating that the latter cannot be calculated but instead needed to be measured experimentally… Pi could be measured experimentally, but it has an exact definition and can be calculated outside of any experiment.
If you run Buffon's needle a lot you come to the conclusion that we live in a locally Euclidean space. That's fine and good. We also live in what you might consider a "locally Newtonian" world, but when things get very big or very small, the Newtonian approximation breaks down.
The ratio of a circle's circumference to its diameter (or equivalently the sum of triangle angles) has the same problem. If you want general relativity to work, then we need to live in a curved spacetime. Depending on whether that spacetime is positively or negatively curved, the angles of a very large triangle may add up to more than or less than pi radians.
Buffon’s needle assumes a flat space, while a non-Euclidean space or geometry would affect the probability leading to a different value other than pi. You can treat the space around us as Euclidean, but that isn’t true for every part of the universe.
Even in a non-Euclidean space with positive or negative curvature, the limit of the ratio of the circumference to the diameter of a circle as the diameter goes to zero is pi.
Most of the article was over my head, but it hit HN because tomorrow is PI Day. It is a nice article though. A Quote:
>we celebrate Good-Enough Pi Day in honor of the approximation 3.1, which is close enough for most purposes. It could be celebrated on either the 3rd of the 1st month ... or on the 1st day of the 3rd month...
at the end it went to the fact in other countries there is no month 14 :)
1 Kings 7:23 describes a basin in Solomon's Temple as ten cubits across and a cord of thirty cubits encircles it. The word for 'cord' has two spellings with two numerical values: qvh (111) and qv (106). Now, 3 × 111/106 = 333/106 = 3.1415
I agree with you on the absolute beauty of e^{i\pi}+1=0, but I think I would also see a different beauty in e^{ik\pi}=1 for all integer k (if we had \pi=6.28... instead)
I would argue that #2 is better with tau; yes there's a factor of 1/2, but it should be there for the same reason that kinetic energy is 1/2mv*2 and distance traveled under constant acceleration is 1/2at*2. All of those are the result of integrating a linear function.
> William Oughtred, in his 1631 work Clavis Mathematicae (The Key of Mathematics), used the notation “π/δ” where π is the circumference of a circle and δ its diameter
This notation would make more sense if you described it as referring to the "perimeter" of a circle, which it seems almost certain is what William Oughtred had in mind. (Though maybe he was thinking "periphery", which is a more exact match to "circumference". I don't know where "circumference" came from; there is no conceptual difference to explain the difference in word use, and it's strange for a Latin word to be used in geometry anyway.)
Aryabhāta, an exceptional Indian mathematician born in 476 CE, left a lasting mark on mathematics and astronomy. At the age of 23, in 499 CE, he calculated pi to be approximately 3.1416 and suggested its irrational nature, relying on insights from Vedic traditions. He’s also widely recognized for introducing zero as a numeral and developing various mathematical and astronomical concepts. That said, the value of pi had been explored even earlier by another Indian mathematician, Baudhayana, around the 6th century BCE. Baudhayana not only worked out pi but also laid out what we now call the Pythagorean Theorem, long before it reached European scholars.
> whereas the Europeans would prefer it not exist.
I (a UKian) use YMD ordered formats everywhere possible and have done for many years, a few decades in fact, perhaps even before I know of ISO8601. I've seen more pushback about it from USians than us EUians, for no reason other than it isn't mm/dd/yyyy. In my experience most people outside technical circles on both sides of the big wet are unaware and just use their local legacy format, when EUians see yyyy-mm-dd (or the undecorated yyyymmdd) they tend to get it and accept it (though often keep using what they have always used), where some very vocal USians take a different format existing as a direct comment that their usual format is wrong and feel that as a personal slight rather than technical matter.
It isn't used on any official forms and such that I'm aware of, at least outside technical circles, but people don't tend to have a problem with YMD ordered formats when they are used.
Right, I thought this was about locale formats and the equivalent of `ls -t`. It didn’t occur to me that the sensicality of using a lexicographic date format in file names might be under contention.
This kind of sloppy writing is too frequently encountered and it is annoying: "Archimedes didn’t think of 22/7, 223/71, or pi as numbers; to him, they were ratios of magnitudes".
This has nothing to do with "thinking". This is just about (incorrect) language translation and mathematical terminology.
In Ancient Greek and Latin and in any other old languages, the word "number" designated the result of a counting operation, i.e. it corresponds to what in modern mathematical terminology is named "natural number", so it must be translated as such in a mathematical context.
In Ancient Greek and Latin, the word "magnitude" (or "measure") designated the result of a measurement operation, like the measurement of a length with a standard ruler or the measurement of a weight with weighing scales, i.e. it corresponds to what in modern mathematical terminology is named "real number", so it must be translated as such in a mathematical context.
There is no difference between Archimedes and a modern mathematician, both distinguish natural numbers and real numbers, but since they speak different languages, they use different words for these 2 concepts. Whenever you read an ancient mathematical text, "number" must be understood as "natural number", while "magnitude" or "measure" must be understood as "real number".
The names do not matter, this is just a problem of language translation (which is done usually incorrectly for texts with mathematical terms, like also for texts with specialized terms from physics, chemistry, biology or mineralogy, because the translators have little knowledge about those domains).
The important difference between ancient mathematics and modern mathematics is that the ancients did not have a method of construction of the real numbers from natural numbers, which has been conceived only in the 19th century, by the time of Cantor, Dedekind etc. The ancients also did not have the method of abstract algebra, where you prove something for a set of axioms of a given form (e.g. of group, of ring, of field etc.) and then the proof is valid for any set where that set of axioms is satisfied. Abstract algebra is also a creation of the 19th century.
Because of that, the ancients had 2 independent sets of axioms for natural numbers and for real numbers, even if many of them were identical in form. The consequence of this was that many theorems and demonstrations had to be duplicated for natural numbers and for real numbers, because something proven for one of them could not be applied directly to the other.
I think there's a sense in which moderns feel that the reals (or at least the rationals) are a natural category: that 5 and 0.3 are the same "kind of thing". Mathematicians talk about the distinction between different classes of numbers, but to most people they're all numbers.
Do you think that sense was shared by the ancients, or do you think that the linguistic distinction mirrored a stronger mental separation between the two? It sounds like it might have done if they had to do the work of duplicating proofs from one to the other. Did they have a single word to describe the shared concept?
To the ancients, as already clearly expressed by Aristotle, natural numbers and real numbers were both quantities. Quantities were classified in discrete quantities, of which natural numbers are an example and continuous quantities, of which real numbers are an example. This classification remains completely valid today.
The ancient mathematicians and philosophers were using the word "quantity" in most contexts where modern people use the word "number", i.e. when the word is applied to different kinds of "numbers", not just to natural numbers.
There is no difference in thinking between ancients and moderns, it is just a difference in the words that happen to be used.
Both the similarities and the differences between natural numbers and real numbers are well entrenched in most natural human languages since many millennia ago, before any scientific theory of quantities, numbers and magnitudes, as exemplified by the similarities and differences between questions like "How many ... do you have?" and "How much ... do you have?".
Actually I consider the ancient usage of the words as more sound than the modern usage. There appears little justification for the modern usage of the word "number" instead of the previous usage of "quantity", except that "number" is a shorter word than "quantity", so the change in terminology is just due to laziness, not to any theoretical reason. However what has been gained by saying "number" instead of "quantity" when the wider sense is intended, has been lost due to the requirement for qualifying "number" as "natural", "real", "integer" etc., when the narrower meaning is intended.
Etymologically, "number" is the result of counting, which real numbers and many other kinds of "numbers" that correspond to continuous quantities are not.
Suppose we designate a unit line segment, and from that we pick out a line segment of (what we now call) length 2 and separately a rectangle of area 3. To "an ancient" (like Aristotle), does it make sense to add the length and the area to get 5? If you then drew a line segment of length 5, would they say it has the same magnitude?
(My understanding is "no" to both questions.)
Right, I read Aristotle's metaphysics, the terms used refer to concepts for what we use a different word, today.
So what you said is very important to take into consideration.
I've read Tau Manifesto [0], and I am now convinced that 3.14... is not the best circle constant, "turn" (tau, τ) with value 6.28... would be a much better choice.
How many radians in full circle? 1 tau (turn). What about 1/3 circle (120 degree)? 1/3 tau. What about euler's constant? e^i*tau=1, or in other words rotate vector by 1 turn and end up at start position.
So beautiful. So unreachable - pi has so much legacy, there is zero chance of changing it.
[0] https://tauday.com/tau-manifesto
There are some advantages also for taking Pi/2 as the circle constant.
The choice between 2xPi and Pi/2 is equivalent with the choice between the cycle and the right angle as the unit of plane angle (the right angle corresponds with the "imaginary" unit, i.e. any point on the unit circle corresponds to i^x, with x in right angles). When extended to 3 dimensions, the corresponding constants for the solid angle become 4xPi and Pi/2, so Pi/2 is more consistent (it corresponds to taking the orthogonal trihedron as the unit of solid angle). The consistency of Pi/2 remains true for higher-dimensional spaces, but that has little practical importance.
"Pi" is by far the worst choice, in all computational applications either 2xPi or Pi/2 are needed, not Pi.
The only case where Pi appears naturally is in measurements, because both the circumference and the diameter are much easier to measure than the radius, and Pi is the relationship between these 2 practical measurements, allowing the conversion of one to the other.
Nowadays, the computational uses of Pi are many orders of magnitude more frequent than the conversions between the practical measurements of diameter and circumference, so the use of Pi is undesirable inside computer programs or in symbolic processing of mathematical formulae.
In general, it is much better to implement trigonometric functions where the argument is either x*Pi/2 or x*2*Pi, instead of traditional trigonometric functions, because the argument reductions are fast and exact. Sadly, the floating-point arithmetic standard defines useless functions of Pi*x, like sinPi, cosPi etc., instead of using any of the 2 better choices.
The only possible advantage of traditional trigonometric functions is at integration or differentiation, where they could save a multiplication, but in almost all applications of this kind the argument of the function is not x, but an expression with at least one multiplication, and the additional multiplication with a constant caused by using x*Pi/2 or x*2*Pi can frequently be done at compile time, or at run time, but only once, not at every computation.
Tau replacing pi? Sure, just start using it. Publish with it. If we want better, we have to make it happen!
I use it in my code at least.
Area of a circle?
Yes, even that one gets more beautiful, too.
Look at the usual equation: A = πr². Why is there no "2" there?
Let's derive it, and in particular, let's derive it from the onion proof, which is that a circle's area is composed of many small circles, arranged concentrically, like a 2D onion:
A = ∫_0^r 2πt dt
There's that blasted 2 again. The tau form is more beautiful:
A = ∫_0^r τt dt
Integrate it, and you'll get A = τr²/2, the constant being a result of the integral.
That is, to me, the usual equation is more properly A = 2πr²/2, the two 2s being different in their origins, and we just usually use & memorize the simplified form.
Unfortunately the ancients didn't invent calculus. Pi had been in use a long time when Liebniz and Newton came along.
The far superior constant, Tau, mentioned. I am glad Mr. Propp has heard the good word.
He's got it wrong. People don't support tau like he supported the Red Sox, the support tau because it's actually a handy constant. Pi still exists!
Tau is not an inconsequential lost cause. I think it will slowly win over everyone.
The real nerds celebrate octal pi day on 3/11. The crowds are too big on 3/14 anyways.
https://imgur.com/a/gczeqkz
The real nerds celebrate Tau day on 6/28.
That's just controlled opposition.
I'm still surprised that Pi isn't considered a universal physical constant, given Buffon's needle depends on (IIUC) the physical curvature of nonrelativistic space.
Is it because pi isn’t measured, but calculated? The wikipedia article (https://en.m.wikipedia.org/wiki/Physical_constant) makes a distinction between a mathematical constant and a physical constant, stating that the latter cannot be calculated but instead needed to be measured experimentally… Pi could be measured experimentally, but it has an exact definition and can be calculated outside of any experiment.
If you run Buffon's needle a lot you come to the conclusion that we live in a locally Euclidean space. That's fine and good. We also live in what you might consider a "locally Newtonian" world, but when things get very big or very small, the Newtonian approximation breaks down.
The ratio of a circle's circumference to its diameter (or equivalently the sum of triangle angles) has the same problem. If you want general relativity to work, then we need to live in a curved spacetime. Depending on whether that spacetime is positively or negatively curved, the angles of a very large triangle may add up to more than or less than pi radians.
Buffon’s needle assumes a flat space, while a non-Euclidean space or geometry would affect the probability leading to a different value other than pi. You can treat the space around us as Euclidean, but that isn’t true for every part of the universe.
Even in a non-Euclidean space with positive or negative curvature, the limit of the ratio of the circumference to the diameter of a circle as the diameter goes to zero is pi.
Up until 2019, it sort of was, via the magnetic constant mu_0 = 4 pi x 10^-7 H/m exactly. Unfortunately the 2019 revision of SI redefined it.
Most of the article was over my head, but it hit HN because tomorrow is PI Day. It is a nice article though. A Quote:
>we celebrate Good-Enough Pi Day in honor of the approximation 3.1, which is close enough for most purposes. It could be celebrated on either the 3rd of the 1st month ... or on the 1st day of the 3rd month...
at the end it went to the fact in other countries there is no month 14 :)
Approximate Pi day is in July: 21/7.
1 Kings 7:23 describes a basin in Solomon's Temple as ten cubits across and a cord of thirty cubits encircles it. The word for 'cord' has two spellings with two numerical values: qvh (111) and qv (106). Now, 3 × 111/106 = 333/106 = 3.1415
Shouldn't that be 22/7?
Looking at the list of 10 equations.
For me 2,5,8,9,10 being simplified with Pi=3.141 is the best outcome, especially 5.
I'm definitely: e^{i \pi} + 1 = 0
I agree with you on the absolute beauty of e^{i\pi}+1=0, but I think I would also see a different beauty in e^{ik\pi}=1 for all integer k (if we had \pi=6.28... instead)
I would argue that #2 is better with tau; yes there's a factor of 1/2, but it should be there for the same reason that kinetic energy is 1/2mv*2 and distance traveled under constant acceleration is 1/2at*2. All of those are the result of integrating a linear function.
For all integers k, e^(i k\tau) = 1
This is way more beautiful to me than the pi version e^(i 2k\pi) = 1
This is the first time I encountered the quarter pi formula:
1 − 1/3 + 1/5 − 1/7 + …
Amazing …
There is a playlist by 3b1b for different interesting derivations of Pi: https://www.youtube.com/watch?v=8GPy_UMV-08&list=PLZHQObOWTQ...
> William Oughtred, in his 1631 work Clavis Mathematicae (The Key of Mathematics), used the notation “π/δ” where π is the circumference of a circle and δ its diameter
This notation would make more sense if you described it as referring to the "perimeter" of a circle, which it seems almost certain is what William Oughtred had in mind. (Though maybe he was thinking "periphery", which is a more exact match to "circumference". I don't know where "circumference" came from; there is no conceptual difference to explain the difference in word use, and it's strange for a Latin word to be used in geometry anyway.)
Aryabhāta, an exceptional Indian mathematician born in 476 CE, left a lasting mark on mathematics and astronomy. At the age of 23, in 499 CE, he calculated pi to be approximately 3.1416 and suggested its irrational nature, relying on insights from Vedic traditions. He’s also widely recognized for introducing zero as a numeral and developing various mathematical and astronomical concepts. That said, the value of pi had been explored even earlier by another Indian mathematician, Baudhayana, around the 6th century BCE. Baudhayana not only worked out pi but also laid out what we now call the Pythagorean Theorem, long before it reached European scholars.
I usually hold that the only sensible date ordering is
yyyy-mm-dd
So that files sort properly.
I'm happy to see that this enabled pi day, whereas the Europeans would prefer it not exist.
> yyyy-mm-dd
> whereas the Europeans would prefer it not exist.
I (a UKian) use YMD ordered formats everywhere possible and have done for many years, a few decades in fact, perhaps even before I know of ISO8601. I've seen more pushback about it from USians than us EUians, for no reason other than it isn't mm/dd/yyyy. In my experience most people outside technical circles on both sides of the big wet are unaware and just use their local legacy format, when EUians see yyyy-mm-dd (or the undecorated yyyymmdd) they tend to get it and accept it (though often keep using what they have always used), where some very vocal USians take a different format existing as a direct comment that their usual format is wrong and feel that as a personal slight rather than technical matter.
ISO 8601 is the standard date format in a number of European countries, but you do you.
It is? I've only ever seen dd/mm/yyyy or dd/mm/yy as standard date format, with different separators instead of / depending on the country.
It's used in Sweden, at least.
Source: Me, having lived there most of my life.
Second source: https://en.wikipedia.org/wiki/Date_and_time_notation_in_Swed...
It isn't used on any official forms and such that I'm aware of, at least outside technical circles, but people don't tend to have a problem with YMD ordered formats when they are used.
Sweden used YYYY-MM-DD before ISO 8601 was even written, and many other countries use it too.
I know at least some countries further East used it as a matter of course, but I wasn't aware any countries in/near Europe did.
YYYY-MM-DD+Z is pretty much the standard.
The Europeans have 22/7, as long as they write dates with the slash.
It is common in Norway to also write 22.7.
File sorting normally isn’t determined by the date format used. You’re probably thinking of other contexts.
Dates in the file names can be sorted by that date if you use the correct date format.
This is common to do.
Right, I thought this was about locale formats and the equivalent of `ls -t`. It didn’t occur to me that the sensicality of using a lexicographic date format in file names might be under contention.