## How did mathematics begin?

Discussions concerned with knowledge of measurement, properties, and relations quantities, theoretical or applied.

Forest_Dump wrote:You are a little confused again. There is an old saying that the bones (i.e., dice) have no memory and so it is here. Hypothetically, you could flip the coin 1000 times and get 1000 heads. It is, in fact, unlikely that you will get 500 of each but certainly possible. If you keep track of the actual record of tries, every conceivable order of heads or tails is equally possible (assuming the coin and tosses are done fairly). Try it. Your could also perhaps more easily just calculate all the potential orders of coin tosses and calculate the number of ways it could be 500-500, 499-501, 501-499, etc. There is no real law at work here other than what you will derive in this way.

I understand that Forest. What I am trying to say is that even if you flip the coin and get heads a thousand times in a row this will eventually even out over time(the odds) if you keep doing this long enough, and Las Vegas is a perfect example of the same odds evening out despite some aberations from time to time.
ronjanec
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Probability is a law of large numbers. It is a pattern that emerge naturally when a large number of events are carried out. Sometimes, pattern is not something that can be seen or obvious. It is hidden in every aspects of live. So, mathematics will be found useful in social sciences too.

Terry
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Terry wrote:..........

Terry,

You define mathematics as “transcendent," and "outside of ourselves."

why do you do this? What is the consequence of doing that?

You're suggesting that mathematical objects have some external reality. Where is that?

I'll say that your perspective/view is a full-fledged theistic position. You have your bias, and I have mine. I'm not a believer, so I do not share this position. I'd rather do math than argue what mathematics actually is.

For months I've been agonizing over Riemann Hypothesis. Do I need to be concerned about the nature of mathematics? No, I don't ... I leave it to others to worry about it.

We're Still debating with Plato
Where do mathematical objects live?

Since Barry Mazur (a mathematician at Harvard University) and Brian Davies (a mathematician at King's College London) are mentioned in the article, here are links to their views:

Please read : Mathematical Platonism and its Opposites: Barry Mazur, a mathematician at Harvard University.

And, article Let Platonism Die on pages 24 and 25
Giacomo
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For convenience I'm posting the article by Brian Davies: Let Platonism Die

Source: http://www.ems-ph.org/journals/newslett ... -06-64.pdf

Over the last few years I have noticed that a number of Fields medallists and other famous mathematicians are being asked by interviewers whether they are Platonists.Many are quite unprepared for this question and try to evade it, or give answers which indicate that they have not thought seriously about it.

Mathematical Platonism comes in many flavours, but two particular elements are usually present. One is the assertion that there exists a mathematical realm outside the confines of space and time in which ideal forms of mathematical entities exist. This should be taken literally – the realm is independent of human society and would exist even if human beings had never evolved.

Theorems are statements about the properties of these mathematical entities, so their truth does not depend on whether anyone has a proof or even of whether there could be a proof (pace Gödel). If you believe that theorems are objectively true before they have been proved, but that mathematics is a creation of human beings in much the same way as music, law and chess are, then you are not a Platonist. I do not want to discuss this aspect of Platonism, about which much has been written,[1,3].

The other aspect of Platonism is that it involves a definite claim about the way the human brain functions. Platonists believe that our understanding of mathematics involves a type of perception of the Platonic realm, and that our brains therefore have the capacity to reach beyond the confi nes of the physical world as currently understood, albeit after a long period of intense concentration. If one does not believe this then the existence of the Platonic realm has literally no significance. This type of claim has more in common with mystical religions than with modern science. This is not surprising, because Platonism grew out of the Pythagorean mystery religion, in which mathematics played a key role.

Although he is a Platonist, Roger Penrose is almost unique in accepting that his beliefs imply that the mathematical brain cannot obey the known laws of physics. His proposals for resolving this problem involve microtubules, and are not generally accepted, [5]. The beliefs of most Platonists are based on gut instincts – strong convictions reinforced by years of immersion in their subject. However, scientifically testable claims are not settled by taking a poll of the opinions of people who have never done any experiments to verify them, even if there is a limited entertainment value observing people reacting to unexpected questions. It seems to have escaped the notice of many Platonists that scientific investigations into the mental processes underlying mathematical understanding are now starting to be carried out. Just as the belief of Kant and many others that Euclidean geometry was the inevitable basis of human thought collapsed, intuitively based claims about how our brains allow us to do mathematics are almost certain to be wrong.

Almost everything that we have learned by scientific experiments about the way our brains operate is not only different from what had previously been thought, but pretty bizarre. One example, related to our geometrical abilities, will have to suffice. Investigation of the brain’s processing of vision show that the image that impinges on our retinas is analyzed in a variety of different ways, into edges of various orientations, colours, etc. which are sent to the brain separately. It then constructs a threedimensional ‘image’ of what the outside world might be like by combining these fragments with other contextual clues, including the memory of the observer. Many types of optical illusion show that this construction can easily fail to match reality. Whether or not an illusion disappears as soon as one realizes that it is one depends on the depth at which it is generated. It is worth mentioning that the investigation of optical illusions is now a subfield of experimental psychology, [4].

The study of our sense of number is in its infancy, but one of the most interesting discoveries is that reasoning about numbers is not a function of general intelligence, [2]. It depends on the successful integration of a number of different modules, or circuits, whose locations in the brain can be identified by using imaging techniques based on measuring oxygen uptake. Numbers below five are recognized using circuits, common to many other animals, that are different from those brought into play by humans for larger numbers. If these circuits are damaged in a stroke, it is quite possible for the person affected to have perfectly normal reasoning powers in all situations not involving numbers, but to be unable to see the distinction between 5 and 8. Dyscalculia is now a recognised disability, and this type has a purely physiological basis.

These studies are proceeding systematically and are beginning to provide a genuine understanding of the basis of our mathematical abilities. They owe nothing to Platonism, whose main function is to contribute a feeling of security in those who are believers. Its other function has been to provide employment for hundreds of philosophers vainly trying to reconcile it with everything we know about the world. It is about time that we recognised that mathematics is not different in type from all our other, equally remarkable, mental skills and ditched the last remnant of this ancient religion.

Giacomo
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Giacomo wrote:Terry,
You define mathematics as “transcendent," and "outside of ourselves."
why do you do this? What is the consequence of doing that?

"For me, what mathematicians do is almost like crafting a sculpture in the image of 'God'. Craftsmanship, for it is a human endeavor and activity. 'God', for the transcendental being to be seeked. You can see both the created and discovered aspects in mathematics. I would say the transcendental realm exists independent of human mathematics."
If mathematics is viewed as a language or representation scheme, I have no question that it is invented. Just see how geometry evolved through the history and still developing today. But it is unlike any other human creations such as music, law and money, we find objectiveness in its inherent structure that everyone can agree upon. Do you have freedom in creating the deep essence of Riemann Hypothesis?

Giacomo wrote:You're suggesting that mathematical objects have some external reality. Where is that?
I'll say that your perspective/view is a full-fledged theistic position. You have your bias, and I have mine. I'm not a believer, so I do not share this position. I'd rather do math than argue what mathematics actually is.

Where can something as formless as probability be? Different people have different line of reasonings and the problem appears fuzzy.

Giacomo wrote:For months I've been agonizing over Riemann Hypothesis. Do I need to be concerned about the nature of mathematics? No, I don't ... I leave it to others to worry about it.

Yes, just as when physicists would like to shut up and calculate.
Terry
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Formalists, Platonists and the intuitionists argue and disagree with one another. But the fact is : the debate never really got resolved.

I found a book online that should be interesting to both of us:

Platonism and anti-Platonism in mathematics by Mark Balaguer

In this book, the author, Mark Balaguer, demonstrates that there are no good arguments for or against mathematical platonism

In Part I, he shows that Platonism is defensible by introducing a novel version of platonism, which he calls full-blooded platonism

In Part II, Balaguer defends anti-platonism
Giacomo
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Giacomo,

Thanks for the good reference information. We can share more about how we do mathematics.

Something funny for our original topic.
Terry
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### Re: How did mathematics begin?

By Terry
“I think the universe has an intangible structure”
I kind of agree but couldn’t the universe be tangible without structure! If by structure your implying some measurable quantity Then I can see the opposite being true, not that I have any authority in these matters, I mean is math’s a revelation or an invention discovered or contrived
I think it is quite plausible that it doesn’t exist outside of the human mind at all, and that these patterns we observe in structures and confirm with what we call math’s are only abstract navigational points, a way of coping with or making sense of, incrementing, dissecting and generalizing about a totally fluid universe with no gaps borders, symmetry’s, center, beginnings or end, in fact if we take time as an example the days are different lengths not just seasonally gravitational friction ensures they will never be exactly the same ever , but that’s not useful in our small existence so we generalize and close enough is good enough but is there structure or is it imagined?

Fuqin
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### Re: How did mathematics begin?

Fuqin wrote:I kind of agree but couldn’t the universe be tangible without structure! ... is there structure or is it imagined?

There are many things that we create mentally and impose on physical entities such as culture and language, moral and law, value and money. Modern society is largely built on the mental infrastructure in our mind which is just as vivid as the physical infrastructure. And we ever go into higher level of abstraction in our mental life. Even scientific theories are mental constructs that we impose on nature whose very nature is independent of our view. We can view gravity as an invisible angel, a Newtonian force or an Einsteinian spacetime wrap and its very own nature has never been changed. We just don't know what it is. We only know what we think it is.

To discover mathematical structures, mathematicians either start with physical entities and abstract to what remain as independent of physical form or directly delve into abstraction by symbolic manuplation without ever care what the symbols really refer to. In the pure abstraction, it is absolutely devoid of physical form. So, we do not refer to any physical structure. It is the pure abstract structure on its own.

For example, when we talk about the abstract numbers, we do not mean the symbols written to represent those abstract numbers. The symbols and number system are human inventions that mirror the Platonic realm of the abstract number. So, by manuplating the symbols, we get an handle on something that we can never touch.
Terry
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### Re: How did mathematics begin?

The Greeks got their maths from the Persians.

http://www.nytimes.com/slideshow/2010/1 ... ef=science
NYCFalcon
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### Re: How did mathematics begin?

NYCFalcon wrote:The Greeks got their maths from the Persians.

http://www.nytimes.com/slideshow/2010/1 ... ef=science

I thought this article was interesting, and it may be evidence that the early Persians had a method which might be seen as proving what they believed true (seeing as how these methods formed part of the instructional setting in which their form of mathematics is revealed). However, insofar as proof in its organized sense is missing, it might not qualify as disputing the claim that the early Greeks invented mathematical proof.

The real question that should be asked here is: what do we mean by the term: 'mathematics' -- or what counts as doing mathematics. Many of the responders seem to think that it requires the usage of abstract ideas, something, I'm assuming humans are capable of, and has to precede its invention.

In thinking about this, I would guess that the earliest mathematics probably involved counting, and possibly the assignment of numbers to a count (i.e., names given to mean a particular count, or possibly a reference to some count, or even an approximate count). I've seen evidence of counting in markings on rocks that seem to relate to the phases of the moon. Other evidence can be found merely by Googling this question. Seeing as how oral language precedes written, possibly by 1000s of years, there is probably no way to actually discern when such counting capability began.

Given this early evidence, though, is it the case that such counts, or even their having names associated with them, qualify it as sufficiently abstracted from their usage to make the claim that it is doing mathematics? (I'm assuming mathematics as a concept has to be distinguished from other activity.) I don't know. I even have a bit of trouble with the early Greek ideas in this area, some of which were confounded in the paradoxes of Zeno. Nonetheless, one would expect that within whatever ages we are speaking about there probably are a few smart individuals around that gravitate to making generalizations about what one is doing, enough to be called mathematicians anyway. Given this, I'd say mathematics might have been invented much longer ago that we have evidence of.

But, because there is undoubtedly a cultural or environmental aspect to this that precludes venturing too far, or, alternatively, provides sufficient freedom of thought to such folks, I'm inclined to think that the early Greeks are the ones who gave birth to the mathematics we know and love today. With respect to the how portion, I'd account for this, as Jay Bronowski does, in the particular environment in which the Greeks are situated where geometry becomes a key factor in measurement of the world it inhabits. much of it involving islands between which navigation becomes important.

James
owleye
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### Re: How did mathematics begin?

owleye wrote:Given this early evidence, though, is it the case that such counts, or even their having names associated with them, qualify it as sufficiently abstracted from their usage to make the claim that it is doing mathematics? (I'm assuming mathematics as a concept has to be distinguished from other activity.) I don't know. I even have a bit of trouble with the early Greek ideas in this area, some of which were confounded in the paradoxes of Zeno. ....
James

mi impression is that until one does not get in trouble with Zeno's paradoxes, they have not met (or invented) mathematics, as that kind of problem only arises when one tries to theorize, to determine rules and mechanisms and criteria to deal with numbers (or with measurements), i.e. well after one gets to simply count things...

neuro
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### Re: How did mathematics begin?

neuro wrote:mi impression is that until one does not get in trouble with Zeno's paradoxes, they have not met (or invented) mathematics, as that kind of problem only arises when one tries to theorize, to determine rules and mechanisms and criteria to deal with numbers (or with measurements), i.e. well after one gets to simply count things...

What I was implying in my remarks about the ancient Greeks in my brief look into the question was not so much about counting, but rather with their understanding of the term: 'continue' (which, as can be noted, I hadn't mentioned). I get the impression that this term is fundamental to their understanding of continuity, one that is used not only within the arguments of Zeno, but is embedded in Euclid's construction proofs and in one or more of his axioms. And I can't help thinking that this term is laden with time. One might say that one 'continues' not only in a spatial sense, but in a temporal sense as well. Geometry, then, can be said to be an applied science, one that deals with space and time -- one that is able to navigate the waterways the Greeks are heavily invested in. Moreover, this way of thinking of continuity seems to carry over well into the time of Newton, and even within Kant's first critique. I should add that he argued for a particular kind of distinction between applied and pure mathematics, a distinction that depends on his dual forms of intuition.

James
owleye
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### Re: How did mathematics begin?

Etymology of mathematics is actually a hobby of mine. I began researching into it because I didn't know any math, and could not understand how any one could make sense of it.
Of course Sumarians were the first civilized peoples to have math but it started in a forest when a cave person made notches on a stick (best we can get to first signs of..), which progressed to early sheep herders. For every sheep a man had as he coraled them in at night he put one pebble into a bag because he didn't know how to count. In the morning when he released his flock, if he had any pebbles left over then that meant either a sheep died, or was stolen (calculus is a greek word meaning pebble.). Some years (100's) later Mesopotamian's began using wet clay to place integers (a single digit in the shape of a flag) on in the sequence of a square to create the number four. A large flag to the left of this formation was given the value of five. This method became costly, and time consuming. So one of the more inventive men made hardened clay tablets, small one for the digit one, and big one's for the denomination of five. This gave way to the abacus (another greek word meaning "tablet".). Now there is some debate which came out first. The abacus, or the number zero. You see the number zero didn't come out until 2k years after math began! The concept of having nothing meant very little to people who usually had nothing.
After meso-method came the greeks. Now there's a tough number system. It is very difficult to add, or subtract in Roman numerals, and I would love to meet the teacher that is willing to take the time in explaining multiplacation, because it is said to be next to impossible.
As for math today (skipping a lot), we owe our decimal (base ten) system to the arabic, and india. The number system we use today was founded by them, and also works wonders with the abacus. The first algorithim was evented by Al'kwarizmi, an Arabian of the Bagdad area while studying at the house of wisdom. The Arabic were also responsible for providing the astrolab. A device used to chart lattitude. The english provided the compass for longitude. And the first computer program was written by Ada Byron Lovelace maybe 100 years before the first computer was even built! I think I'm out of room here?
nahtahni
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### Re: How did mathematics begin?

nahtahni wrote: I think I'm out of room here?

Actually, you can't run out of room. Posts can be arbitrarily long. But thanks for the excellent post and history lesson!

linford86
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### Re: How did mathematics begin?

nahtahni wrote:The first algorithm was invented by Al'kwarizmi, an Arabian of the Bagdad area while studying at the house of wisdom.

This is inaccurate.

Scholars commonly credit the first recorded algorithm to Pāṇini, who lived almost a millennium before al-Khwārizmī. Pāṇini's algorithms are not useful for calculation; instead they present a way of parsing well formed Sanskrit grammar. Noam Chomsky regards Pāṇini the inventor of the first generative grammar, and a forerunner to modern theoretical linguistics.

Other algorithms existed prior to al-Khwārizmī's work. Famous algorithms include Euclid's algorithm for finding greatest common divisors, the Sieve of Eratosthenes for producing tables of prime numbers, the algorithm underlying Sun Zi's constructive proof of the Chinese Remainder Theorem, and Heron's Method for approximating square roots. It's worth noting that Heron's method dates back to ancient Babylon - when it was actually invented is not entirely known.

The connection of al-Khwārizmī to algorithms is as follows: the word "algorithm" is derived from the Latin Algoritmi de numero Indorum, which was what scholars commonly called the untitled Latin translation of al-Khwārizmī's text on the Indian numbering system. This Latin title means "al-Khwārizmī on the Hindu Art of Reckoning" [source: wikipedia]. al-Khwārizmī never provides a name for the mechanical numerical manipulations he invents in his manuscript.

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