Mathematical Truths: Discovered or Invented?

[zoobyshoe]

No one claimed you could, zooby. Math is a language that is designed to describe specific and limited things.

Or, maybe it's not a language.

What I perceive happening here is the same thing that happened with the "Physics doesn't do 'why' questions" meme that infected PF for a while. Someone makes a forceful or otherwise attention-getting statement that doesn't get properly disputed in a timely way, and it takes off and starts having a life of its own. Everyone repeats it as if it's gospel, doctrine, and starts rationalizing their own support for it.

Zooby, do you at least recognize that your claim here is non-mainstream? That math is generally regarded to be a language?

Citations? I'm going to need a slew of them to support the claim of "generally regarded".

In the meantime here are some definitions of math, none of which characterizes it as a language:

Definitions of mathematics
Main article: Definitions of mathematics
Aristotle defined mathematics as "the science of quantity", and this definition prevailed until the 18th century.[29] Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[30] Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals.[7] There is not even consensus on whether mathematics is an art or a science.[8] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[7] Some just say, "Mathematics is what mathematicians do."[7]

Three leading types of definition of mathematics are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought.[31] All have severe problems, none has widespread acceptance, and no reconciliation seems possible.[31]

An early definition of mathematics in terms of logic was Benjamin Peirce's "the science that draws necessary conclusions" (1870).[32] In the Principia Mathematica,Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic" (1903).[33]

Intuitionist definitions, developing from the philosophy of mathematician L.E.J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other."[31] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems".[34] A formal system is a set of symbols, or tokens, and some rules telling how the tokens may be combined into formulas. In formal systems, the word axiomhas a special meaning, different from the ordinary meaning of "a self-evident truth". In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

-wiki

noun
1.
(used with a singular verb) the systematic treatment of magnitude,relationships between figures and forms, and relations betweenquantities expressed symbolically.

http://dictionary.reference.com/browse/mathematics

: the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations

http://www.merriam-webster.com/dictionary/mathematics

1. (Mathematics) (functioning as singular) a group of related sciences, including algebra, geometry, andcalculus, concerned with the study of number, quantity, shape, and space and their interrelationships by usinga specialized notation

http://www.thefreedictionary.com/mathematics

noun The definition of mathematics is the study of the sciences of numbers, quantities, geometry and forms.

http://www.yourdictionary.com/mathematics

The word "science" crops up most. I think math as language is the concept that's not mainstream. Math is not a communication system.

From a previous post:

Math is the language of physics. That's why physics laws are nothing more than mathematical relations.

No, the language of physics is the large system of very specifically defined terms, a few of which I mentioned earlier. The jargon of physics. Physics employs math as its main tool, to keep track of quantities. That's not communication, it's accounting, as Feynman (sort of) put it. And, physics laws are not "nothing more than" mathematical relations. The physics in F=ma lies in the hard won grasp of the concepts of force, mass, and acceleration. The mathematical relation, as such, a = bc is no revelation. The formula great meaning derives from the concepts that were found to have that simple relation. As I have said a few times on the forum, Newton's laws were the result of about 2000 years of struggle to get traction on the phenomenon of motion. It was not at all apparent how to parse the situation. Starting from scratch and defining mass, for example, in the presence of the confounding phenomenon of weight was not easy. Figuring that, and the other concepts, out is the physics. Once that was done, the mathematically trivial relationship eventually became apparent. For every success like Newton's 3 laws, there are uncounted failures that aren't in the books. We cherry pick the things we've found a hand hold on and forget about the failed attempts that lead nowhere. Physics is about figuring out what to account for.

Saying mathematical truths are discovered is like saying chess truths are discovered. Both statements ignore the fact you're making discoveries about a human mental invention and falsely imply you're making discoveries about nature.

Does that mean the the universe didn't know how to make objects move properly until Galileo(?)discovered f=ma?

F=ma is a physics concept, arrived at by experiment and observation. It's not a math concept. We didn't learn simple multiplication from accelerating masses. Multiplication was invented to make repeated addition easy and fast.

I didn't claim we did.

Yes you did, indirectly. You were denying math was invented and claiming it was discovered in nature. Therefore, we must have learned multiplication from nature, perhaps by observing F=ma, or some such. Your point was some vague claim math has always existed in nature before we 'discovered' it, was it not?

Yes, multiplication was invented to make repeated addition of quantities of objects found in reality easier.

Yes, and not to "describe what's around you". To avoid getting ripped off at the market. All the early evidence is that early arithmetic was commerce driven; accounting. But, at least you're now admitting it was invented.

I'm really surprised that you don't seem to appreciate that math doesn't get cooking until you abstract it from real world representation and deal with quantities as quantities. That's where all the advances are made. Read Euclid. There's no hint of a mention in the whole book about practical applications. The ancient mathematicians worked forward fascinated by the logic in and of itself, with no particular concern whether or not there might be a mundane use for any of it.

For Newton's laws of motion (in particular, gravity and orbits), Calculus was invented to help describe them.

Again, glad your referring to it as "invented."

 

[zoobyshoe]

Aside from the self-evident that f=ma is an equation and an equation is a mathematical statement ... and without knowing its history of how it came to be all you would know is that f=ma is a purely logical mathematical proposition (so, yes, f=ma is math), what I think you're really quibbling with here is whether f=ma is true in nature and/or can be called a Truth.

F=ma is not a mathematical statement. It is a physics statement. The variables stand for variable quantities of specific physical phenomena, not abstract quantities in general. It's a statement about the relationship of force, mass, and acceleration. There's no point in invoking someone who doesn't know that, we all do. We know the intent of it is to describe nature, not abstract number relations.

This is a math statement:

(a + b)² = a² + 2ab + b²

It holds true for any two numbers a and b.This is a physics statement:

V = IR

Only holds true when V = voltage, I = current, R = resistance. It does not hold true when V = 20, I = 100, R = 365. It has no significance, conveys no meaning when divorced from the proper physics concepts.

 

[Pythagorean]

@zoobyshoe Mathematics is not defined as a language, it's language status is a matter of classification. Looking up definitions and not finding "is a language" is essentially an appeal to ignorance, but I'll gladly provide you with citations.

It's very common in science and mathematics to talk about math as a language, in the classroom, in research meetings, etc - I say this anecdotally as someone who participates in academia as a researcher, student, and teaching assistant.

Anyway, here are some citations:

http://www.cut-the-knot.org/language/MathIsLanguage.shtml

http://www.ascd.org/publications/books/105137/chapters/Mathematics-as-Language.aspx

https://www.dpmms.cam.ac.uk/~wtg10/grammar.pdf

http://www.tandfonline.com/doi/abs/10.1080/00131720008984764?journalCode=utef20#.VFuO__nF8qI

http://www.jstor.org/discover/10.2307/20205297?uid=3739448&uid=2&uid=3737720&uid=4&sid=21104975748287

 

[zoobyshoe]

@zoobyshoe Mathematics is not defined as a language, it's language status is a matter of classification.

All the definitions I posted encompass classifications. Math was most often classified in them as a science, then as a field of study. None classified it as a language.

It's very common in science and mathematics to talk about math as a language, in the classroom, in research meetings, etc - I say this anecdotally as someone who participates in academia as a researcher, student, and teaching assistant.

Anyway, here are some citations:

http://www.cut-the-knot.org/language/MathIsLanguage.shtml

http://www.ascd.org/publications/books/105137/chapters/Mathematics-as-Language.aspx

https://www.dpmms.cam.ac.uk/~wtg10/grammar.pdf

http://www.tandfonline.com/doi/abs/10.1080/00131720008984764?journalCode=utef20#.VFuO__nF8qI

http://www.jstor.org/discover/10.2307/20205297?uid=3739448&uid=2&uid=3737720&uid=4&sid=21104975748287

Your anecdotal report and the citations are some support for Russ' assertion that it is "generally regarded" as a language by people involved in math. Which is what I requested.

However, on reading the citations, I find a lot of assertion without support, assertion with incomplete support, assertion with incompetent support, and generally, a confused shift from speaking about the 'language of mathematics' (its jargon) to 'mathematics is a language.'

The first link is the worst, having the largest amount of inarticulate puffery. The links with more serious attempts to get people to regard math as a language are misguided attempts at trying to formalize an informal "manner of speaking": the way mathematical concepts are expressed is extremely specific and rigorous, and that can be off-putting. You have to enter into the spirit of it to make sense of it, and you might informally encourage someone to do that by suggesting they consider it a separate language unto itself, until such time as they get the hang of it. However, it looks like that heuristic is being misunderstood as a fact, and the links show linguistics being scoured for possible similarities between language and math, in the attempt to make a case by confirmation bias, as in the last link. There's no point to this. All you need to do it advise students they have to pay particular and sustained attention to terms and concepts.

 

[collinsmark]

I'm in the camp that mathematics is a language. But it's more than that.

The Embalse Nuclear Power Station in Argentina, a pressurized heavy water (PHWR), is a building. Yes, Embalse is a building, but it's more than just a building. Similarly, mathematics is a language, but it's more than that.

Mathematics is a language, but it's a very special, very precise language. It is also a tool of logic (tool being the key word here). And in addition to those, it is an embodiment -- an abstract repository so to speak -- of a particular body of knowledge: the very logic itself of previously proven ideas.

Language:
I think it was Stephen Hawking that related an anecdote told to him by his publishers when writing one of his lay-person targeted books. They told him something to the effect that every equation he put in his book would drop the book's sales by 10% (or some-such). I guess most lay-people are scared away by equations and will put the book down if they see one.

I find that sad, because communicating certain ideas are so much clearer in my opinion if they are communicated with equations and math. One could say, "the gravitational force between two bodies is proportional to the area of a rectangle who's length takes on the mass of one body and who's width takes on the mass of the other body; and the force is also inversely proportional to the area of a square who's sides are the distance between the two bodies." Gah! Are you kidding me? Just say,
F = G \frac{m_1 m_2}{r^2}.
That's so much more clear to me, and communicates the same idea! Then again I've taken the time to learn what that notation means, with the multiplication and the division and the squaring operation. So I understand that maybe some others less well versed in mathematical notation would prefer the former. If only they knew what they're missing!

Tool:
Egill Skallagrímsson, born in Iceland in the tenth century, in a time and place where the common person could not read a written language (paper and parchment had not been introduced to Scandinavia yet), was a famous warrior-poet. While most people couldn't read or write, Egill on the other hand had learned the "runes." He could read and write. I can imagine Egill scratching out a poem-in-progress in the dust by his feet as he's working out the details, "Hmm, that doesn't quite fit. Let me scribble that out. Ah, that word works much better here." A written language can be a wonderful tool to organize one's thoughts before speaking them.

English is a spoken language but also has a written form, and they are different from each other. Although the written form does not have the pronunciation, and the spoken form does not have the spelling, nor the same detail of punctuation. (Before arguing that the written language simply imitates the spoken form, ask yourself why the language has words with silent letters.) Try to read a book to someone out loud, and the style is noticeably different than it would be if the story was created and spoken on the spot. It becomes quite obvious that there is more to the written word than just the fact that it is written.

This is where the written form of mathematics really shines. It's able to communicate ideas and relationships that simply couldn't be clearly spoken. It aids one in organizing one's thoughts in ways that the common, non-mathematical language fails.

With only a little effort, starting with
F = G \frac{m_1 m_2}{r^2},
I can use mathematics to reorganize that idea and say,
r = \pm \sqrt{\frac{G m_1 m_2}{F}}.
That would be difficult if the only language available was English.

[Edit: And elaborating on this tool idea, sometimes when physicists and engineers use mathematics to rearrange and combine thoughts it can help lead to new, unexpected ideas. For example, when Paul Dirac reformulated the ideas of non-relativistic quantum mechanics with the principles of special relativity, it lead to hints of antimatter. The existence of antimatter "fell out of the math" so to speak (this relates to the \pm sign when you take the square root). Mathematics is not just a tool, it can be used, in part, as a predictive tool.]

Knowledge:
Pythagoras proved that for a right triangle, c^2 = a^2 + b^2. Brahmagupta derived the roots of a second order polynomial to be x = \frac{-b \ \pm \sqrt{b^2 - 4ac}}{2a}. Euler showed us that e^{ix} = \cos x + i \sin x (and yes, this is a provable relationship -- not a mere definition or "trick"). Once proven, those ideas become added to the overall body of mathematical knowledge. We don't need to re-prove them from scratch when working on other things; we can leverage the results and go from there.

So mathematics isn't just the language of logic, it also embodies the very logic itself of previously proven theorems.

---------------------------------------------------
I'll end this post with a love poem I wrote several years ago:

A love poem, by collinsmark:

The number of ounces per ton,
less a dozen times square fifty-one,
with the cube of neg-nine
together combine
to make three score, less one to the none.
​

32000 -(12)(51)^2 + (-9)^3 = 3(20) - 1^0