Math: Did we discover or create it?

[Pyrotex]

Originally Posted by Qfwfq

It isn't contra-reality, no. I'ts in contradiction with Peano's axioms. It's also false in the various Z_n groups. OTOH, 4 + 8 = 3 is false in N but true in Z_9.

Okay. In abstract math perhaps.

I guess when I use "reality" I am talking physics. In those formulas which scientists use to describe physical reality, it is rare to find an integer. There are some squares and cubes, but those define something you DO to the term; you multiply it times itself.

I dare to say that nowhere in physical formulas do you find numbers (real or integer) where the base of that number is of any importance or consequence at all. The formulas that are composed of variables (or numeric constants such as pi*) do NOT assume or even need a particular numeric base.

We can generalize this to: ALL physical formula which define or describe the natural world are INDEPENDENT of numeric base; the truths they represent do not depend on using a particular base.

Even in college algebra, one learns that the properties of mathematical functions are independent of base. A=B ; B=C ; therefore A=C. This works in ALL bases. The same for AB = BA, in those algebras where multiplication [what's the word?!] conjugates {?}.

*pi can be expressed in any base, or as the abstract ratio of circumference to radius of a circle.

[Bio-Hazard]

Everything in the universe exists, yet math is abstract. It was built upon more with the discoveries and potentials it held that were undiscovered. Altered, formed, built upon. A never ending line of discoveries are held to be connected to its foundation.

[questor]

math is a language. it started simply and evolved into the myriad and complex language it is today. it is a means of communication and due to it's
truth and purity, it can be spoken and understood by any nationality in the world.

[Pyrotex]

Originally Posted by questor

math is a language. it started simply and evolved into the myriad and complex language it is today. it is a means of communication and due to it's
truth and purity, it can be spoken and understood by any nationality in the world.

After all is said and done (mostly said, not so much done), I still think we have a syntactical problem.
"Math is a language" is True.
"French is a language" is True.
But the word "language" in the first sentence does NOT have the same meaning or usage as the word "language" in the second sentence.

[CraigD]

Originally Posted by Pyrotex

"Math is a language" is True.
"French is a language" is True.
But the word "language" in the first sentence does NOT have the same meaning or usage as the word "language" in the second sentence.

This difficulty is one of the key observations of the early 20th century discipline of General Semantics. Although GS is a complicated discipline, it’s easy to summarize how it addresses the problem of the same word used for 2 different things: it “subscripts” them. Thus, in GS, we’d write:
"Math is a language1";
"French is a language2",
and keep adding to our table of precise definitions of the words usage as new and distinct ones are used.

That few people outside of academic Sociology have ever heard of GS suggest the influence it has had on 20th century Western society.

[virtualmeet]

Originally Posted by Pyrotex

After all is said and done (mostly said, not so much done), I still think we have a syntactical problem.
"Math is a language" is True.
"French is a language" is True.
But the word "language" in the first sentence does NOT have the same meaning or usage as the word "language" in the second sentence.

from the cambridge dictionaries :
Language : A system of communication consisting of sounds, words and grammar, or the system of communication used by the people of a particular country or profession.
I can't see anything in this definition that contradict saying that mathematics is a language...

[Pyrotex]

Originally Posted by CraigD

This difficulty is one of the key observations of the early 20th century discipline of General Semantics. Although GS is a complicated discipline, it’s easy to summarize how it addresses the problem of the same word used for 2 different things: it “subscripts” them. Thus, in GS, we’d write:
"Math is a language1";
"French is a language2",
and keep adding to our table of precise definitions of the words usage as new and distinct ones are used.

It is a privilege to meet someone cogent of General Semantics. Thank you. Your solution is perfect, after adding:

language1 is NOT equal to language2

[Pyrotex]

Originally Posted by virtualmeet

from the cambridge dictionaries :
Language : A system of communication consisting of sounds, words and grammar, or the system of communication used by the people of a particular country or profession. I can't see anything in this definition that contradict saying that mathematics is a language...

Going to the dictionary to define what YOU mean in your argument is usually a good thing, commendable, and avoids confusion.
Going to the dictionary to (re-) define what someone else means in their argument is fraught with peril.

The definition you presented is a good one and it is "correct" as far as it goes. But the fallacy lies in presenting a broad, general definition and concluding that all "things" that fall within that definition are "equal". Not so.

Let us take the word "language" as an example. French is a language. The word for 'shit' in French is 'merde' -- pronounced in a particular way. Where did that sound or pronunciation come from? Did the nature of feces itself <<determine>> the word that the French would choose or how to pronounce it? Of course not. Has 'merde' always been pronounced the same through the centuries? We have reason to believe it has not. The sounds for each and every word in French (and English) are arbitrary. Indeed, even the way the words are used is arbitrary, custom, tradition, changing.

Algebra is a language. There is no word for 'shit' in algebra. What the HELL?!?! Why not? It's a 'language' isn't it? We can translate French into English ('shit') so why can't we translate French into Algebra?

Algebra is a language. Where are its verbs? One might say 'add', 'subtract', etc. Perhaps variables are pronouns and numbers are nouns. But you can only take this comparison just so far and it breaks down in the specifics. Algebra is NOT a language like French is a language. The point here is that we are now NOT using 'language' in its broadest sense, but are using 'language' in very narrow, specific senses.

And how about 'add', 'subtract', and the other operators. Are they arbitrary? Have these processes changed over the centuries with custom and tradition? No. Not in the same sense as the pronunciation, spelling and common usage for 'shit' or 'merde'. Don't go there. Is the way addition performed arbitrary in the same way that the spelling for 'merde' is essentially arbitrary? No. Did the nature of addition itself <<determine>> the operator symbolized by '+'? Absolutely. Has the way we 'add' evolved over the generations? No. Adding doesn't evolve, but spoken languages do.

These are just a few of dozens of differences we could point out between Algebra and French. And they are a different KIND of difference than you find between English and French. You cannot go to Paris and find someone in the bookstore who 'speaks Algebra' and carry on a conversation about your children and pets in Algebra, no matter how good you are at factoring quadratic equations.

English and French = language1
Algebra = language2
language1 is NOT = language2

Does this answer your question?

[Doctordick]

I thought I would jump in here as I am quite disappointed with level of the discussion on a rather simple subject. First, you are all talking about opinions and not facts and there is no recognition of that fact to speak of. As I have commented elsewhere, dictionaries can not be used to learn the meanings of words in a language as the meanings of the words used in the definition of a word must be known before the definition can be used to understand the word. We actually learn what words mean by guessing their meaning from usage and, for that reason, every language extant is vague and imprecise when examined carefully. (Which is also true of what I am saying here by the way, and does much to explain Goldwater's famous gaff, "listen to what I mean, not what I say!")

As an aside, the fact that all languages are vague and imprecise plays a very important role in the advancement of human thought because it allows misunderstanding and misunderstanding can lead to new perspectives which might better explain things than what was intended by the source of the explanation. If our perspective on reality never changed, we would still be living in caves. (An opinion by the way and not put forth as an argument.)

And Pyrotex, your comments are perhaps the best in the thread but you also are misrepresenting some important issues. (Again, an opinion only )

Originally Posted by Pyrotex

Going to the dictionary to (re-) define what someone else means in their argument is fraught with peril.

Dictionaries are a stabilizing factor in communications and your comment smacks of Goldwater's gaff.

Originally Posted by Pyrotex

Algebra is a language. There is no word for 'shit' in algebra. What the HELL?!?! Why not? It's a 'language' isn't it? We can translate French into English ('shit') so why can't we translate French into Algebra?

No one said all languages are equal and this can often lead to communication problems. The fact that we "walk in space" is due to exactly such a problem. The Russians were the first to do it and when they announced their accomplishment they used a word which does not exist in English. In Russian they have two different words for "to go", one to go in a vehicle and the other to go without a vehicle. When translated into English "to go without a vehicle" is commonly translated as walk, but it could just as reasonably be translated as swim or fly or slide for that matter. I also understand Chinese has no word for "freedom" and is translated to a word meaning something closer to "out of control". (Of course, as you said, meanings change and evolve and the meaning of that word might very well be different to the next generation of Chinese.)

Languages evolve to handle the subjects of interest to the users of that language and there exist many differences between many languages. I very much doubt the Pirahã have a word for snow. The important concept here is that "languages" are used to communicate and that includes communication with ourselves (it is common knowledge that it is very difficult to consciously think about things you have no words for).

Originally Posted by Pyrotex

And how about 'add', 'subtract', and the other operators. Are they arbitrary? Have these processes changed over the centuries with custom and tradition? No. Not in the same sense as the pronunciation, spelling and common usage for 'shit' or 'merde'. Don't go there.

Here you are confusing concepts with the symbols used for the concepts. In mathematics, particular care has been made to assure that the total collection of concepts are internally consistent thus impeding misunderstanding and slowing any evolution of the referenced concepts. "+" means "to add" and "to add" is a rather universally well understood and useful concept (except, it seems, in the Pirahã language). Now the verb "to add" in English has evolved to include things such as attractiveness or health (very imprecisely defined concepts) because they want to talk about these issues. The verb "to add" in mathematics has also evolved to include things like modular addition but the concept is quite precisely defined (otherwise it wouldn't be in mathematics).

Originally Posted by Pyrotex

These are just a few of dozens of differences we could point out between Algebra and French. And they are a different KIND of difference than you find between English and French. You cannot go to Paris and find someone in the bookstore who 'speaks Algebra' and carry on a conversation about your children and pets in Algebra, no matter how good you are at factoring quadratic equations.

They are not a different KIND at all; they all have to do with concerns of the people who developed the language. You couldn't carry on a conversation about your children and pets in "Algebra" anywhere because these issues are insufficiently well defined to be included in "Algebra"; no more than you could have a discussion about snow in Pirahãnese. Notice that I invented that word! The rules of English are sufficiently imprecise that such invention can take place under the "looks like a usage I have seen before so he'll probably guess what I mean correctly" rule. You can't do that in mathematics; you must be very careful to assure the meaning is well understood before you can use a symbol "that's what we call what stands for "words" in mathematics".

Sorry about that, but I had just expected a little more precision in your thoughts than I found here.

Have fun – Dick

[Pyrotex]

Originally Posted by Doctordick

I thought I would jump in here as I am quite disappointed with level of the discussion on a rather simple subject. ...As I have commented elsewhere, dictionaries can not be used to learn the meanings of words in a language as the meanings of the words used in the definition of a word must be known before the definition can be used to understand the word. We actually learn what words mean by guessing their meaning from usage and, for that reason, every language extant is vague and imprecise when examined carefully. ...And Pyrotex, your comments are perhaps the best in the thread but you also are misrepresenting some important issues. ...Dictionaries are a stabilizing factor in communications...Dick

I have read carefully. Twice. And I have to agree with 88.3% of everything you said. Approximately.
Seriously, yes, I posted without sufficient cogitation on the metaphor of Algebra as a "language". I rambled. Criticism well taken. Mea culpa.

As to the dictionary, though. Yes, it is an incredibly stabilizing factor, when well and thoughfully used. However, in several threads hereabouts, people have attempted to use a definition out of the dictionary that was soooo broad and general, that it included (at first glance) and number of non-too-related concepts. They then concluded that at least two of those concepts were "identical" since they both fit the definition.

This is not using the dictionary to stablize.

This is using the dictionary as artillery.