As an interesting aside, I'm just getting ready to finish out the semester covering functional programming languages using Racket, which is a close descendent of LISP.

Why would you cause such brain damage.
"Unfortunately a Russian spy stole the last meg of a LISP program for controlling our nuclear defense systems. Fortunately, it was all right-parenthesis."

 

Dare I presume that all this levity is to avoid the original question, which still stands.

I see that no-one has taken me up on comparing a 'Field' in Maths and Physics, or offered other examples.

 

It may not come as a surprise, but would it really be suspect if it weren't a never ending number? Consider the ratio of the radius of a circuit to the length of the sides of an inscribed hexagon. That comes out not only to a rational number, but to an integer, namely 1. Is that suspect?

I didn't say that it would be suspect if it weren't a never ending number; I said it would be as suspect. The difference being that there are as many real numbers that are never ending as those that have a finite number of digits.

 

Dare I presume that all this levity is to avoid the original question, which still stands.

Fair enough... let's get back to topic, then...

The truth is that I have been giving some thought to your question, and I've come to the conclusion that I'm probably not qualified enough to give a satisfactory answer, since my knowledge of mathematics is limited compared to my knowledge of physics and engineering. Although there are more practical fields out there that make heavy use of mathematics other than physics and engineering... economics for instance...
I'd probably revisit the work for Kurt Gödel and have a look at his incompleteness theorem and see if I could draw an example of what you're looking for... don't really know why, it's just a hunch...

 

Just got back from checking the father and son mathematicians... I was mistaken... of course it wasn't Gauss, It was the Bernoulli family I was thinking about... it's fascinating reading.

 

I didn't say that it would be suspect if it weren't a never ending number; I said it would be as suspect. The difference being that there are as many real numbers that are never ending as those that have a finite number of digits.

Ah, this leads to an interesting subtle point. Are you drawing a distinction only between "real numbers that are never ending" and "real numbers that have a finite number of digits"?

This is important because many rational numbers are never ending in a particular number base. 1/3 is never ending in base ten but has a finite number of digits in base-3. 1/10 has a finite number of digits in base ten but is never ending in base-2. I think what you mean to say is "irrational numbers", which have a an infinite representation in any (finite) number base.

And there are infinitely more irrational numbers than there are rational numbers (not just, "as many").

 

Dare I presume that all this levity is to avoid the original question, which still stands.

I see that no-one has taken me up on comparing a 'Field' in Maths and Physics, or offered other examples.

Is the term "field" in mathematics defined to be the same (or embody the same concepts) as the term "field" in physics? Or is it a matter of two branches of knowledge using the same term for conceptually different things?

If the former, then comparing them is a reasonable thing to do. If the latter, then comparing them is fundamentally nonsensical.

In this case, the latter applies. In physics a field is simply a physical measure that has a value at any given point and time within some region of space-time. They could have called it "bob" instead of "field". In math, at least in abstract algebra, a field is the name given to a set of values that obeys certain mathematical properties. They could have called it "sue" instead of "field" and the choice of one group calling it "field" instead of "bob" or "sue" has no impact on the choice that the other group can or can't make.

You might as well as people to compare "Field" in Physics to "Field" in Agriculture.

 

Ah, this leads to an interesting subtle point. Are you drawing a distinction only between "real numbers that are never ending" and "real numbers that have a finite number of digits"?

This is important because many rational numbers are never ending in a particular number base. 1/3 is never ending in base ten but has a finite number of digits in base-3. 1/10 has a finite number of digits in base ten but is never ending in base-2. I think what you mean to say is "irrational numbers", which have a an infinite representation in any (finite) number base.

And there are infinitely more irrational numbers than there are rational numbers (not just, "as many").

I was making the distinction as you describe, not taking into account the more general case of "any (finite) number base." Interesting twist to the problem.

I am struggling with what you were saying but "instinctively" see my error. Let ℝ =(the set of real numbers with a finite number of digits). Is ℝ countably infinite (I think so)? And irrational numbers are not countable and hence their cardinality is larger. Are you saying that |ℝ|=

? And does that mean that |irrational numbers|=

? Or am I going too far?

This was one of my favorite topics in Applied Math at Georgia Tech, and I seem to have foggy memories of it now

 

Is the term "field" in mathematics defined to be the same (or embody the same concepts) as the term "field" in physics? Or is it a matter of two branches of knowledge using the same term for conceptually different things?
If the former, then comparing them is a reasonable thing to do. If the latter, then comparing them is fundamentally nonsensical.
In this case, the latter applies. In physics a field is simply a physical measure that has a value at any given point and time within some region of space-time. They could have called it "bob" instead of "field". In math, at least in abstract algebra, a field is the name given to a set of values that obeys certain mathematical properties. They could have called it "sue" instead of "field" and the choice of one group calling it "field" instead of "bob" or "sue" has no impact on the choice that the other group can or can't make.
You might as well as people to compare "Field" in Physics to "Field" in Agriculture.

With the greatest respect, you seem to have presumed a great deal more about my motives for asking this question than I gave out in the first place.

I do not teach either mathematics or physics or anything else, so I do not confuse people with conflicting definitions.

In the case of the word Field there are indeed conflicting definitions that confuse many students, who may well be called bob or sue themselves.

Much applied mathematics is based on linear algebra which is about vector spaces, V, defined over a field F.
Often that field is the field of real or complex numbers and the elements are what physicists refer to as scalars.
Meanwhile the physicists refer to a spatial diagram or layout of equipollent (contour) lines or little arrows as a 'Field' , which setup does not conform to the mathematical definition of a field.

 

I'm not too up on the terms and, particularly, the symbology used in Abstract Algebra, so I'm not in a position to really answer your quite reasonable questions.

Yes, the set of all real numbers with a finite number of digits would be countably infinite because, if nothing else, you could multiply them by a suitable integer, namely B^n where B is the number base and n is the number of fractional digits, to map them to an integer.

But that extends to a larger set than just the set of real numbers with a finite number of digits since it would also apply to the union of that set and also the set of all real numbers that have an infinite number of digits but that have a repeating pattern, such as you get from 2/3 or 4/7 or, in general, any rational fraction of integers that does not produce a terminating value. So the set of rational numbers is countably finite. But the set of irrational numbers in uncountably finite. I've seen the proofs for this, but it has been a very long time.

 

I am struggling with what you were saying but "instinctively" see my error. Let ℝ =(the set of real numbers with a finite number of digits). Is ℝ countably infinite (I think so)? And irrational numbers are not countable and hence their cardinality is larger. Are you saying that |ℝ|=

? And does that mean that |irrational numbers|=

? Or am I going too far?

Yes, your set R is included in Q the set of all rational numbers (which includes some infinite digit numbers) which is countably infinite.

 

With the greatest respect, you seem to have presumed a great deal more about my motives for asking this question than I gave out in the first place.

I do not teach either mathematics or physics or anything else, so I do not confuse people with conflicting definitions.

In the case of the word Field there are indeed conflicting definitions that confuse many students, who may well be called bob or sue themselves.

Much applied mathematics is based on linear algebra which is about vector spaces, V, defined over a field F.
Often that field is the field of real or complex numbers and the elements are what physicists refer to as scalars.
Meanwhile the physicists refer to a spatial diagram or layout of equipollent (contour) lines or little arrows as a 'Field' , which setup does not conform to the mathematical definition of a field.

I'm not trying to presume anything about your motives -- I simply can't figure out what's the point of asking for a comparison between two things that have virtually nothing in common except that two groups of people coincidentally chose to use the same name for them.

There's more of a basis of comparison for the term "shell" as used in Marine Biology and in Combat Artillery.

That terms have conflicting definitions is not new and it is not unique to math and physics. In my research work at the academy we ran headlong into much more serious contradictions because two groups used the same terms for very different concepts and both groups thought that they were using them to mean the same thing. In this case, the culprit was the concept of a "key" in computer science (cryptography in particular) and electrical engineering (communications in particular). Both sides talk about secure communications but both sides assume that the other side has solved the part of the problem that they don't know how to deal with. The computer science folks know how to use asymmetric cryptography to do away with the key distribution problem associated with symmetric keys and then just say, "and so you use a public key infrastructure in conjunction with a spread spectrum communications system that is intrinsically jam resistant". The electrical engineering folks say, "and so you use spread spectrum communications in conjunction with cryptographic protocols to distribute the transmission security keys". The problem is that the transmission security keys ARE symmetric keys! But the comp sci folks don't know that spread spectrum uses symmetric keys and the electrical folks don't know what a symmetric key is and why distributing them is an intractable problem.

This was such an oft-repeated issue that we actually wrote a paper about it entitled, "Impediments to Systems Thinking: Communities Separated by a Common Language"

 

Loved your comparison of marines, army, sailors and airmen. Great paper.

 

Loved your comparison of marines, army, sailors and airmen. Great paper.

Thanks. It was actually a "junk" paper in the sense that it was very peripheral to what we were doing. Had I not been at a point that I needed to get a "lead author" credit under my belt it probably would have never been written. But writing it was very useful, even had we never submitted it to a conference, because it did force us to really think about the problems were having -- and how to mitigate them -- in communicating our work to others.

The conference that it was submitted to, CITSA, is a truly junk conference. They will accept anything, even though they claim to be peer-reviewed. But they also tout how their peer-review process reflects "diversity of rigor in thought", whatever the hell that is. Most conferences will not publish your paper in the proceedings unless you physically present your work at the conference. Not so with this one. It is run as a "mega conference" in conjunction with several other conferences all run by the same company. I went to the conference taking it seriously only to discover that the overwhelming number of authors never bothered to show up. I didn't attend a single session that even had half of the authors present and a couple sessions didn't have a single author there to present their work. Of the ones that did, most probably shouldn't have -- there was a lot of really flawed work that was presented. I originally went there rather embarrassed about what I was going to present and came away realizing it was by far one of the better papers there -- which said far more about the quality of the other papers, for sure.

 

Thanks. It was actually a "junk" paper in the sense that it was very peripheral to what we were doing. Had I not been at a point that I needed to get a "lead author" credit under my belt it probably would have never been written. But writing it was very useful, even had we never submitted it to a conference, because it did force us to really think about the problems were having -- and how to mitigate them -- in communicating our work to others.

Well, it was great to me to see how well it explained the problems of communication between groups that use narrow technical terms within their groups. Coming from a cryptographic background the word 'key[ing]' has a specific meaning that a data systems expert might believe was similar to a index of a database or as you explain a communications expert might think of as the secure frequency hop sequence in a spread spectrum system that allows for anti-jam or stealth capabilities but is not cryptographically strong as a means to transfer keying material.

 

It's not even a matter of being cryptographically strong. IF both sides already possess a shared key that is secure, then you can transmit anything, including future transmission security keys, in a cryptographically secure fashion without too much difficult. The question is how do you get both sides to have that initial shared secret key? The underlying problem is that both sides think that the other side has the problem solved and that neither side understands that the other side thinks that way. I remember a briefing we made to the #3 person at the NSA (okay, one of the #3 people at the NSA) who thought that we were making a mountain out of a molehill because, paraphrasing, "the transmission of cryptographic keys over the air is done all the time using jam-resistant spread-spectrum radio links." To which we responded, "But are you aware that all existing forms of spread spectrum rely on symmetric keys for their jam-resistance?" At which point her eyes got real wide and she said, "Please tell me you're not serious." She, and most others that don't know how spread spectrum actually works, think that the jam-resistance is intrinsic in the waveform and are completely unaware that a key of any kind is involved at all. We went through an almost identical experience with one of the head people at DISA.

 

I been gone for a long time out of the field but we had only a few systems (none cleared for TS) that were cleared for remote key distribution via the same encrypted link as that link might be compromised and we would be giving out the next key in the chain. Key distribution (Symmetric-key algorithms) was always a problem and subject to compromise (Walkers) even with the limited networks we ran back then but PKC asymmetric cryptography type systems only found limited use for audio scrambling devices where information was Sensitive but Unclassified. I'm sure today the compute power needed for just about everything (like a STU-III) is easily contained in a small module instead of a rack.

 

Yep. The example that we generally used was a pretty superficial comparison between the situation with a STU and with systems like HAVE QUICK and SINCGARS. With a STU you have a pretty jam resistant physical layer intrinsically available but you don't with the others.

 

studiot......surely you jest.

Let's take the word engineering out of your title. Please.

There is no need to disparage a noble profession.

Mathematicians and physicists are the same thing.

They believe that the mysteries of reality can be explained by math.

They believe that a mathematical possibility has a natural analog.

Math has never caused anything, because math has no meaning.

Math is an universal tool.....like language. It has no meaning or cause. Does a wrench have meaning?

Is has worth....like any tool.....but how many people do you see use tools the right way?

Every term has to be given physical context.

The problem is....once you put the physical term in an equation......

the mathematical context overrides the physical context.

Maxwell's equations are a prime example.

This practice has compounded since.

But with the state of the world today, this might be a blessing.

I would not trust such knowledge to the arrogant enlightened.

 

Science is an method for studying reality whereas mathematics is not limited to reality. Math is reasoning about quantity, space, structure and change and as such is not empirical or a science. For example, in mathematics it makes sense to talk about any number of dimensions, even negative dimensions or fractional dimensions. Math only requires that it is logically consistent, not that it fits any part of reality.

Sometimes you will hear discussion about how mathematics is invented, whereas science is discovered. This is exactly what they are talking about.