What Constitutes a Tortured Derivation or Proof in Mathematics?

[Shubee]

I'm in the middle of an argument with a physicist. I'm arguing that section 3 of Einstein's famous paper, On the Electrodynamics of Moving Bodies [1], is a tortured derivation of the Lorentz transformation and that the derivation by Professor Rick Field, University of Florida, in his online lecture notes for Physics 3063, doesn't have that tortured quality. [2]. Thus I need a practical definition from a professional mathematician. What constitutes a tortured derivation or proof in mathematics? Any help will be sincerely appreciated.

 

Shubee

1. On the Electrodynamics of Moving Bodies

2. Physics 3063 Spring 2005

3. The Axiomatization of Physics - Step 1

[CraigD]

Thus I need a practical definition from a professional mathematician.

Though my profession is computer programming, I have a 1982 BS in Math, pro consulting experience in survey design and statistics, several years experience in tutoring undergraduate math, and 1 year of teaching remedial college Math, I could arguably be considered a “professional mathematician”, and try to keep up with a fraction of the literature, so I’ll presume to address your request.

What constitutes a tortured derivation or proof in mathematics?

Other than your posts, Shubee, I’ve never heard “tortured” applied to formal math, and find it puzzling. There are certainly counterintuitive derivations and proofs, especially computer generated ones that arguably are altogether beyond the domain of human intuition, and there are demonstrably derivations and proofs that have more algebraic steps or less intuitively obvious ones, a derivation or proof is either formally correct, or not. The idea that one could be “tortured” – forcibly coerced, subjected to inhumane treatment and pain – doesn’t make sense to me.

 

Specific mathematical functions have been described with adjectives such as “ugly” and “pathological”, but this is just an intuitive description intended to help the reader visualize them, much like the adjectives “spikey” and “smooth”.

 

Any impression of torture experienced while reading math is, IMHO, present in the mind of the reader, not an innate quality of the math. Although being able to present math in a way that a wide or a specific target audience can understand and enjoy is a valuable skill in a math teacher or popularizer, it’s not, I think, a critical one for a mathematician communicating with other mathematicians or mathematical scientists.

 

:( Based on my current understanding of math, I recommend not attempting to criticize or categorize math and mathematical science in terms of tortuousness. :hyper: I’m curious, though, Schubee, to see your attempt at a formal definition of “tortured math”, or pointed examples of math you consider tortured, and equivalent math you consider not tortured?

[LaurieAG]

Hi Schubee and CraigD,

 

I'm not a professional mathematician but I did get a distinction average in my higher maths subjects for my B.App.Sc.

 

I’ve never heard “tortured” applied to formal math, and find it puzzling.

 

'Twisted' might be a more appropriate word that could also equally apply to the plot of the Lorentz transformation itself.

 

I would define 'tortured' to the process where calculus is applied, say to integrate/differentiate in relation to time and then integrate/differentiate in relation to another variable (particularly if this other variable comes about as an intermediate state due to the partial differentation based on time).

 

I'll read the articles and get back.

[Shubee]

I’m curious, though, Schubee, to see your attempt at a formal definition of “tortured math”, or pointed examples of math you consider tortured, and equivalent math you consider not tortured?

 

Hi CraigD. I didn't have a formal definition in mind when I wrote the opening post but I did list the examples you request. I suggested that the first link takes you to a tortured derivation whereas the derivations in the second and third link have an elegant quality to them. I now see what the definition should be. Let me know if you approve. A tortured derivation or proof in mathematics is just about any argument that is unnecessarily long and terribly inelegant or just pointlessly longwinded.

[Nootropic]

Some proofs are not quite as elegant as we would like in mathematics. Take for instance Louie De Brange's proof of the Bieberbach conjecture. It was initially impenetrable because of the techniques he used were largely developed by himself (See De Branges Space). His proof, after being reviewed, was revised and shortened (much to his dislike), to fit with our taste for elegance. Mathematics is about problem-solving, but mathematicians have a strong taste for aesthetics.

 

I would define 'tortured' to the process where calculus is applied, say to integrate/differentiate in relation to time and then integrate/differentiate in relation to another variable (particularly if this other variable comes about as an intermediate state due to the partial differentation based on time).

 

Most likely not. Applying calculus doesn't get a special name. Using calculus when other methods are available though is considered bad manners.

[LaurieAG]

Most likely not. Applying calculus doesn't get a special name. Using calculus when other methods are available though is considered bad manners.

 

Like what, the stake?

 

Calculus - Wikipedia, the free encyclopedia

 

Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. Historically, it was sometimes referred to as "the calculus of infinitesimals", but that usage is seldom seen today. Most basically, calculus is the study of change, in the same way that geometry is the study of space.

 

Calculus has widespread applications in science and engineering and is used to solve problems for which algebra alone is insufficient. Calculus builds on algebra, trigonometry, and analytic geometry and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus. In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions.

[Nootropic]

Well, it's not necessarily just calculus. Using methods of proof that are more advanced is usually considered not as simple and elegant. If a proof that is not considered elementary is discovered, mathematicians usually still search for a proof that is considered elementary.

 

Though Calculus is definitely useful, I'm well aware of that and the fact that there are problems that require it. Though there are a number of problems that seemingly require calculus, some do not need it. However, it still comes in handy.

[LaurieAG]

Though Calculus is definitely useful, I'm well aware of that and the fact that there are problems that require it. Though there are a number of problems that seemingly require calculus, some do not need it. However, it still comes in handy.

 

Hi Nootropic,

 

One aspect of calculus you might not be aware of is the repeating differential/integral cycle of 4 different states for the imaginary unit i. Calculus even covers quantum aspects.

[Nootropic]

One aspect of calculus you might not be aware of is the repeating differential/integral cycle of 4 different states for the imaginary unit i

 

This has nothing to do with calculus. It is simply a property of the imaginary unit i.

 

Calculus even covers quantum aspects

 

I am certainly aware that Quantum mechanics makes use of physics, if that's what you meant. However, if you're talking about quantum calculus, I'm also aware of that, but quantum calculus really came from a need in quantum mechanics.

[LaurieAG]

This has nothing to do with calculus. It is simply a property of the imaginary unit i.

 

Hi Nootropic,

 

Considering that the intermediate state of partial differential time is necessary to blend 'space-time', surely the behaviour of the imaginary unit itself and any 'certainty' expected as a result of further 'pure calculus' manipulation (quantum or otherwise) depends on its partial integral/differential level and this repeating cycle.

 

I would define 'tortured' to the process where calculus is applied, say to integrate/differentiate in relation to time and then integrate/differentiate in relation to another variable (particularly if this other variable comes about as an intermediate state due to the partial differentation based on time).

[Nootropic]

Considering that the intermediate state of partial differential time is necessary to blend 'space-time', surely the behaviour of the imaginary unit itself and any 'certainty' expected as a result of further 'pure calculus' manipulation (quantum or otherwise) depends on its partial integral/differential level and this repeating cycle.

 

I still fail to see how this has anything at all to do with calculus. The fact that i does not have distinct powers for every integer really only has to do with the way that is defined, that is, as the square root of -1. This fact was discovered long before calculus ever came into existence.

[CraigD]

The conventional meaning of “calculus” is math involving the limits of infinite series, so I’d say the imaginary unit [math]i = \sqrt{-1}[/math], while not being essential to calculus, is strongly related to it. Likewise, [math]i[/math], though very useful in calculus, has meaning outside of it.

 

A cool quality of [math]i[/math] not found in its real number sibling [math]1[/math] is that it and the operation of multiplication alone can generate an interesting set:

[math] \{x : x=i^n, n \,\mbox{is an integer}\} = \{ -1, i, -i, 1 \} [/math];

or, written less formally, [math]i \cdot i = -1, -1 \cdot i = -i, -i \cdot i = 1, 1 \cdot i = i, \dots[/math].

 

I think this is to what Laurie is referring by

One aspect of calculus you might not be aware of is the repeating differential/integral cycle of 4 different states for the imaginary unit i. Calculus even covers quantum aspects.

Interestingly (to me, at least :)) is that [math]-i[/math] behaves similarly to [math]-1[/math], ie:

[math] \{x : x=(-1)^n, n \,\mbox{is an integer}\} = \{ -1, 1 \} [/math]

and

[math] \{x : x=(-i)^n, n \,\mbox{is an integer}\} = \{ -i, 1 \} [/math]

 

Graphically, these qualities look something like this:

 

This periodic behavior makes for wonderful and useful formulae, like

[math]e^{i \theta} = \cos \theta + I \sin \theta[/math], which relates exponential series to the complex plane, and from which trig function approximations like

[math] \sin X \approx X - \frac{X^3}{6} +\frac{X^5}{120} -\frac{X^7}{5040} +\frac{X^9}{362880}[/math]

which are used by most software and hardware electronic calculators trig features.

 

Pretty elegant stuff, in my eyes.

 

What’s long struck me as deep and mysterious is why there’re not a hyper-complex spaces – that is, why no one seems to have imagined functions of complex numbers that result in other i-like identity numbers. Mathematical reality seems to have an ingrained two-ness to it, if you follow my philosophical drift.

[Nootropic]

The conventional meaning of “calculus” is math involving the limits of infinite series, so I’d say the imaginary unit i = sqrt{-1}, while not being essential to calculus, is strongly related to it. Likewise, i, though very useful in calculus, has meaning outside of it.

 

Most definitely. I don't deny the existence of subjects such as complex analysis. One thing I find particularly interesting is that if a complex function is differentiable at a point, then it is differentiable everywhere. So much easier than those pesky real-valued functions!

 

A cool quality of i not found in its real number sibling 1 is that it and the operation of multiplication alone can generate an interesting set:

{x : x=i^n, n ,mbox{is an integer}} = { -1, i, -i, 1 };

or, written less formally, i cdot i = -1, -1 cdot i = -i, -i cdot i = 1, 1 cdot i = i, dots.

 

A very cool multiplicative subgroup of the quarternions! Which is, of course, isomorphic to the nonzero elements of the integers mod 5!