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Happy Tau Day

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Re: Happy Tau Day

#121

Post by MrManatee »

jeffwass wrote: post #51
MrManatee wrote: post #42

I have found the idea of tau useful even though I have never used it in writing. One argument in favor of tau is that in many formulas pi often has the multiplier 2 in front of it. If these formulas are written in terms of tau, they may become slightly easier to memorize and manipulate. Perhaps so, but I don't really care about this. It’s not a big difference. Besides, there are also lots of formulas that are easier…

Your Cauchy argument is actually misleading and runs right into my joking example elsewhere in this thread about needing to define a new constant Sigma=4pi to work better with steradians. Pi doesn't always represent only a pure circle. Eg, in solid angles there are 4pi steradians over a sphere. Or 2tau. Or what I define as Sigma. This extra factor of two when using Tau should be just as disconcerting to tau enthusias…

As you already realized, my example was supposed to be in 2D, but I forgot to mention it. Sorry about that.

But your point about steradians is still spot on. If I had been talking about the 3D case, there would have been a 1/(2pi) factor in the front of the pdf. Using tau would not have helped anyone realize that this particular 2pi is most naturally thought of as 1/2 of the full 4pi steradians. Using tau is not a substitute for careful thinking.

And I am not even actually advocating switching to tau; I am not sure if it is worth the effort. But I still think that tau is a good way of conveying the idea that the current definition of pi as basically a historical accident. This can be enlightening even if one doesn't start using tau. As a more extreme example, I think it is useful to recognize that the decimal base (as opposed to binary, hex, etc.) is a historical accident, but I wouldn't advocate changing it.

Re: Happy Tau Day

#122

Post by Bromskloss »

JadeNB wrote: post #101
skrebbel wrote: post #30

If mathematics has a bikeshed, this is it. I enjoy the ridiculousness of it all, but people who consider this anything other than a well-executed joke really should get a hold of themselves.

> If mathematics has a bikeshed, this is it. In my experience, few professional mathematicians care about this. The question that Bromskloss ( https://news.ycombinator.com/item?id=11993750 ) mentions, of whether or not 0 is a natural number, is likely to incur a far more passionate response. (At the risk of downvotes: it is.)

In therms of practical consequences such as making mathematical expressions look different, it seems to me that 0 vs. 1 makes a smaller difference than does τ vs. π, because many statements about the set ℕ of natural numbers make no mention of what its first element is anyway. Often you can drag along any of the two definitions of ℕ through the argument, without changing it's conclusions. For example, if f:ℕ→ℝ (and ε∈ℝ, N∈ℕ, n∈ℕ), the limit L∈ℝ is given by (∀ε>0)(∃N)(∀n≥N)(|f(n)-L|<ε), regardless of whether one takes ℕ={0,1,2,…} or ℕ={1,2,3,…}.

Re: Happy Tau Day

#123

Post by formula1 »

devishard wrote: post #39
formula1 wrote: post #31

Earlier quoted context omitted.

I understand this perspective if arguments were a limited resource and that by not fighting about tau we would win other battles (which is not true). The reality is that there is no such thing as time "better spent" withoit enforcing some sort of oppressive regieme and ways to curb imagination. I believe the tau fight is one of the rare good fights.

Arguments are a limited resource. I have 45 minutes on the train and I'm using it to argue about tau, and I don't have the time or energy for more. And when society is divided on a lot of issues, people run into outrage fatigue and become apathetic. And to say that there is no such thing as time better spent unless you enforce an oppressive regime is a perfect solution fallacy. Sure, people will always do inefficient…

Ooooo.... I feel your passion! Great points.

Internet conversations will only take things so far, after conversation generally action is necessary. Whats nice about hacker news is that with many actions we can take action with functional programming or framework xyz immediately. With something like tau, publicity is its problem. So few people know about it that any sort of exposure benefits it. To say that untyped systems are worse than typed requires minimal exposure, simply more experience to understand why

Re: Happy Tau Day

#124

Post by JadeNB »

Bromskloss wrote: post #122
JadeNB wrote: post #101

Earlier quoted context omitted.

> If mathematics has a bikeshed, this is it. In my experience, few professional mathematicians care about this. The question that Bromskloss ( https://news.ycombinator.com/item?id=11993750 ) mentions, of whether or not 0 is a natural number, is likely to incur a far more passionate response. (At the risk of downvotes: it is.)

In therms of practical consequences such as making mathematical expressions look different, it seems to me that 0 vs. 1 makes a smaller difference than does τ vs. π, because many statements about the set ℕ of natural numbers make no mention of what its first element is anyway. Often you can drag along any of the two definitions of ℕ through the argument, without changing it's conclusions. For example, if f:ℕ→ℝ (and ε…

Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`. In that case, it's easy just to say what you mean; but, then again, so is it easy to say what you mean in any occurrence of `\pi` or `\tau`. I spoke only of the passion that the issue evokes, not its practical consequences.

Re: Happy Tau Day

#125

Post by Bromskloss »

JadeNB wrote: post #124
Bromskloss wrote: post #122

Earlier quoted context omitted.

In therms of practical consequences such as making mathematical expressions look different, it seems to me that 0 vs. 1 makes a smaller difference than does τ vs. π, because many statements about the set ℕ of natural numbers make no mention of what its first element is anyway. Often you can drag along any of the two definitions of ℕ through the argument, without changing it's conclusions. For example, if f:ℕ→ℝ (and ε…

Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`. In that case, it's easy just to say what you mean; but, then again, so is it easy to say what you mean in any occurrence of `\pi` or `\tau`. I spoke only of the passion that the issue evokes, not its practical consequences.

> Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`.

Certainly. To be clear, I don't mean to argue against anything you said. I just find it to be an interesting observation that statements quite often are true for both versions of ℕ.

Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numbers, as captured by the Peano axioms. https://en.wikipedia.org/wiki/Peano_axioms#Formulation

Re: Happy Tau Day

#126

Post by JadeNB »

Bromskloss wrote: post #125
JadeNB wrote: post #124

Earlier quoted context omitted.

Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`. In that case, it's easy just to say what you mean; but, then again, so is it easy to say what you mean in any occurrence of `\pi` or `\tau`. I spoke only of the passion that the issue evokes, not its practical consequences.

> Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`. Certainly. To be clear, I don't mean to argue against anything you said. I just find it to be an interesting observation that statements quite often are true for both versions of ℕ. Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numb…

> Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numbers, as captured by the Peano axioms.

I find this statement interesting and plausible, but difficult to formalise. The Peano axioms explicitly name the least element of the natural numbers, so in some sense they foreground, rather than effacing, the issue of whether it is 0 or 1; but, on the other hand, I suppose that one could argue that an argument using "only the most fundamental essence of the natural numbers" is one that, as it were, is allowed only to use a constant naming the least element, without pre-supposing any idea of what that name 'means' in any external axiomatic or intuitive sense.

Re: Happy Tau Day

#127

Post by Bromskloss »

JadeNB wrote: post #126
Bromskloss wrote: post #125

Earlier quoted context omitted.

> Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`. Certainly. To be clear, I don't mean to argue against anything you said. I just find it to be an interesting observation that statements quite often are true for both versions of ℕ. Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numb…

> Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numbers, as captured by the Peano axioms. I find this statement interesting and plausible, but difficult to formalise. The Peano axioms explicitly name the least element of the natural numbers, so in some sense they foreground, rather than effacing, the issue of whether it is 0 or 1; b…

I think I follow, and I agree. How I view the natural numbers on this fundamental level – and I would say it is the picture that the Peano axioms nail down – is as consisting of a first element and then an infinite chain of elements after it (equivalently: the natural numbers are a chain that extends infinitely in one direction but not in the other). As you say, it does not matter, then, if that first element is called "0", "1", or "17" (until we bestow additional structure on the numbers by introducing addition and multiplication operators).

Re: Happy Tau Day

#128

Post by wtbob »

hexane360 wrote: post #92
wtbob wrote: post #43

Earlier quoted context omitted.

> millions of Americans use imperial measurements There are two types of countries: those using 'metric' units, and those who have been to the Moon. There's absolutely nothing fundamentally wrong with standard units (indeed, they are better for concrete manipulation). One can do science and engineering just as well with grains as with grams, with cups as with litres, with inches as with centimetres. They could do wit…

But I think the standard unit system is one of many reasons engineers rely on formulas for 100% of their calculations. When you have that many conversion factors just to get between energy, force, distance, power, etc., you lose out on the universality of physics. At that point you find someone who has already done all the unit conversions and isolated it in a nice factor out front. Also, IMO stuff in decimal is almo…

> Also, IMO stuff in decimal is almost always easier to compare than fractional. When someone asks me for a size up from an 8mm wrench, I know to grab the 9mm. When someone asks me for a size up from 5/16", it'll take me a bit to get to 11/32".

A lot of that is just due to how we teach fractions — but decimal notation really is nice. I wish that we used duodecimal instead: all the advantages of decimal, but with a far better base.

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