Og has been on a group hunt. The proceeds are to be divided equally (or otherwise). Og observes the dismembered limbs of various creatures and concludes that someone has already taken something, since the limbs cannot be sorted into pairs.
A capacity to find cheats (or to cheat) puts food on the table.
As an economist, I am rarely said to have the soul of a poet, so thanks. I am frequently told to get a life though.
“Developed” seems to fit better than discovered or invented. The fundamentals of math have always existed; much earlier than the first time Thag the caveman tried to figure out how big a wooly mammoth he needed to kill to feed all the guests his wife invited for dinner and cocktails.
The 90 degree angle, the linear equation, calculus, trig, etc. have been around since time began; it’s just that no one had much of a use for them or understanding of the math arts until humanity had to begin developing what amounts to “accounting” for stuff (no, I am not a math whiz) fairly recently.
I’d suspect the earliest rules for “math” were based on addition and first understood by Thag; “Honey, how many days did that last wooly last? How many clams do I need to dig up and bring home today?”
Time passes and understandings are developed as needed.
Please excuse if this example has been used before, but the thread was simply too darn long to read in its entirety.
I would say that man “discovered” certain mathematical properties and “invented” math to predict or describe them.
An analogy is that man discovered time and invented clocks to keep track of it.
I have just types this once and lost it all, please forgive me if it seems a little rushed this second time.
There is none.
I can describe exactly a circle or other mathematical construct to an infinite degree.
The ‘Laws’ of Physics can try but you will find that they are not accurate to an infinite degree.
You may not be able to find me a tangible circle (or other construct) But when you do I will be able to describe it exactly. In the future I may be able to tell you more about it, but what I have now will still be true, * exactly*.
This is where Physics and Maths differ, Physics is a model, some would say a language, used to describe the universe. It does so but not exactly. You can not predict the orbits of the planets exactly, many ‘Laws’ are built on observation (although many are derived mathematically from previous laws - along this line if a law is derived mathematically, as long as what it is derived from hold true the law will always be true)
With Maths we can describe exactly the concepts exactly. This is the difference Maths is not just a language it is much more than that (and someday physics may achieve this). It defines the very nature of the universe, it does not attempt to explain gravity or space time. But the concepts that it describes will hold true forever to an infinite degree. This is also why it is discovered. For any circle our mathematically descriptions will be true, for any construct.
You may not be able to find me a tangible circle, or even an exact ellipse but when you do I guarantee I will be able to describe it, you may say this is because invented it. I say this ‘If Maths was invented how can we build on it in the way that we do, how does it link so well together why do pi and e and i show up in so many places (and in nature too)’.
Are we inventing these If so we must be very good.
I may discover/prove a rule that describes something - this rule may include pi, or any other mathematical ideas. I didn’t know that it would when I started but it does. Am I a very good inventor? or not?
You know this post was much better the first time round, oh well.
kid_gilligan’s analogy isn’t bad, but where some of us disagree is on whether the clock in question “includes” time or “describes” time. Specifically, whether properties such as ordinality and cardinality, which have natural expressions, are sufficient to declare mathematics is discovered.
I submit that they are not. THoughts have a natural expression. Whether a human ever expresses those thoughts in auditory code to another human being is irrelevant to the existence of thought. Nevertheless, language was invented not discovered. All useful languages describe objects whose existence is independent of the constructed language. Most useful languages also describe elements whose existence is entirely dependent upon the constructed language. Mathematics is a very useful set of languages.
For those arguing that mathematics is not a set of languages, please tell me what elements of a language mathematics does not possess.
Gartog
You can define a circle or any other mathematical construct. Your definition will be “exact” because it is tautological.
In other words, if an object happens by which conforms exactly to your definition, then your definition will be perfectly correct. Obviously. I similarly can predict the future with absolute accuracy. It may not be your future, but if you find the person whose future it is, my prediction will be exactly correct.
What, exactly?
No. It defines the constructs of mathematics. Physics attempts to apply these constructs to the nature of the Universe. That application still does not define nature, it describes it.
Consider your run-on clause. How can mathematics “define” the Universe without gravity? The Universe we observe manifestly contains gravity.
We can build upon maths because they have been continually refined, improved, and formalized. It was worth the effort for mathematicians to do this because maths are very useful for describing reality. Pi and e show up so often in our observances of nature because they describe ratios (or coonvergencies) which are often found in nature. 8,345,522,454,547,876 shows up less often in our observances because it does not correspond to a perceptibly common ratio or convergence. Nouns show up very often in our observance of nature. The subjunctive case shows up less often. Neither is a good argument for the “discovery” of English.
awesome thread Psi Cop!
I only regret that I have discovered it a week late.
Hopefully I won’t trudge too heavily on points that have been made already.
I think that the dichotomy of “invent” and “discover” is false in this case. Math is an invention which allows discoveries. Yes there beautiful leaps of imagination and problem solving which are inventive such as Riemann geometry, but the usefulness of these inventions is usually judged by what can be discovered through the invention. In other words, how interesting are the conclusions reached from the initial postulates?
Is pi still pi before or after it’s value has been calculated to some digit? Yes, but only if you accept the invention of the circle with the properties of a radius and a circumference.
Euclidean geometry provided an amazing amount of insight into the physical world even though its conception of points and lines don’t exist in nature. Similarly, the invention or conception of numbers separate from physical objects or quantities allowed the discovery that 2+2=4 etc…
You know, I have been a regular visitor to the Straight Dope site for several months now, and whenever I drop by, I usually check out the threadspotting section. But this is the first topic that got me interested enough to actually register so I could join in.
As a mathematician by education, programmer by occupation, and linguist by hobby, I am forced to agree with those who say math is discovered and the language of math is invented. (I have only read the first page of this thread, so I’m sorry if I repeat information contained in the second page.)
The ancient Egyptians, the ancient Chinese, and the ancient inhabitants of India all discovered certain properties of right angles, certain properties of prime numbers, and pi. (Admittedly their values for pi were not as precise as ours.) Right angles, prime numbers, and (ideal) circles may all exist only as abstract concepts, but when three very different and physically separated cultures all come up with the same abstract concepts and work out the same properties for working with those concepts, it makes me think that those abtract concepts have a reality independent of the specific minds that first thought them, especially when those concepts become useful in modern celestial mechanics, which the those minds had no concept of at all.
Perhaps that reality exists only within the human mind (i.e., the human mind in general, not the mind of a specific human), but in that case mathematics has existed since the first human mind came into being, and is discovered, not invented. If these things are products of how humans perceive the world, then they are hardwired into our minds and they are discovered as properties of the human mind rather than as properties of nature, but they are still discovered. I personally do not believe that they are properties of the mind only, but the only way to settle the question (short of the heavens opening up and deity revealing the answer) would be to meet an intelligent alien species that perceived things in a different way and had a system analagous to our mathematics that was nonsensical to us but actually worked.
Another example of independent discovery of mathematical principles would be Newton and Leibniz with calculus. But I suspect that some on this list would say they were simply discovering unknown properties of a system that had previously been invented.
As further support for the idea that mathematics exists as a property of the human mind in general (again, I do not believe this is the case, but I believe it is possible), Noam Chomsky postulated that all humans have the innate ability to learn language. This part of our mind is termed a language acquisition device, or LAD. While most of Chomskyan linguistics is no longer widely accepted, the concept of an LAD is still considered valid by most linguists (with the exception of behavioralists). Now, If mathematics is only an abtract construct, then it must exist in all human minds (due to the example I provided above), and just as we use the LAD to learn a language, we use an analagous ability to learn the language of math. Language helps us to express thoughts, the language of math helps us to express mathematical concepts.
However, I believe that mathematics is as real as gravity or mass; that is, it is a property of the physical world.