Active Now

Element 99
Randy D
Discussion » Statements » Rosie's Corner » Some math seems incredibly unwieldy. Was math INVENTED or was it DISCOVERED? Anyone?

Some math seems incredibly unwieldy. Was math INVENTED or was it DISCOVERED? Anyone?

Posted - June 4


  • 3083
    Hmmm! I think that's an old argument and I don't recall anyone giving a very definitive answer.

    You could say that simple counting is artificial so arithmetic and maths are invented, and certainly all measurement units are arbitrary to some extent at least.

    Yet if you analyse natural process, many undeniably work to numerically consistent rules, so those rules are discoveries not inventions.

    If we work back from that, we can find that mathematical relationships are themselves consistent, but we have discovered rather than invented them.

    You can test the point very easily with a bowl of 12 apples: share them among 4 people and each person has 3 apples. It don't matter what you call "three" and "four" - the words are just our language - nor how you perform the sharing, but the effect is the same.

    The BBC Radio Four statistics-magazine programme More Or Less recently posed a simple question: what is 48 plus 27 (or similar numbers). Solve by mental arithmetic. This set the "soshul-meejah" world a-flutter as users twigged the point being made, and explained in a later edition, was not just the right answer (75) but how do you reach it? Many respondents said they round one or both numbers to the nearest decade then add or subtract the roundings from that sum. It showed there is not always a single right or wrong approach in arithmetic and maths, but the right approach is one that gives the right answer.  Again this is not really an invention but a lot of individual "discoveries" of techniques.

    Go up a stage and consider (a+b)(a+b): it will always equal [a^2+2ab+b^2]. You can test that by replacing the a and b with simple numbers like 2 and 3: it always works. It gives different answers with different values but the rule is consistent.

    The Ancient Greek geometricians who analysed plane figures like the triangle did not "invent" their Theorems and Proofs. They measured lots of triangles and discovered the relationships and rules by which plane geometry works.

    Same with pi: no-one invented 3.1416.... It just happens to be true whatever units of measure you use, and in whatever word-language; and very simple to prove for yourself with a fabric tape-measure and assorted cylindrical objects!

    [Incidentally there is a peculiar verse in the Old Testament that famously says that as the columns in King Solomon's Temple were 1 cubit in diameter they were "exactly" 3 cubits round. I've wondered about that - was the story written before the discovery or widespread knowledge of pi, and slavishly copied by later translators who might have spotted the error? I think it is talking about gilding the columns, but I am sure the gilders would have done what you or I would do: wrap the gold leaf round the column and cut it off at the overlap!]

    So is Maths invented? Well, 12/4 = 3 is abstract, having no physical embodiment except as symbols. Sharing those 12 apples around 4 people is physical objects giving the abstract a concrete form and no oranges were involved, only apples. Yet 12/4=3 would be just as true if within some abstruse calculation from astro-physics showing how a certain star works. Could anyone be said to have "invented" the notion that twelve divided by four is three? That a 1-cubit diameter column is 3.142 cubits round? That the common logarithm of 1 000 000 is 6? Or that people found these questions' concepts, and that they work whatever names our language give the values?

    So I would say Mathematics is a matter of discoveries, but of fundamentally abstract concepts, patterns and relationships rather than physical objects.  
      June 7, 2021 5:59 PM MDT

  • 108905
    Thank you for your thoughtful informative and detailed reply m'dear. I don't understand a lot if but I don't have to. I take your word for it. I was thinking though. Maybe everything was always already "THERE". And our lives are spent "discovering" them. Like a many-room house with all the doors closed. Over time you go through the house opening doors and "discovering" what lies behind them? There is nothing abstract about 12 apples divided evenly among 3 people. The physical proof ot truth is right there in front of you. It's the other stuff that goes above and beyond. Pi for instance. You can't touch it, consume it or use it to purchase things. So 1 plus 1 is always going to equal two no matter what you do. One of me and one of you has got to equal two human beings. When does it not? That higher math that can cover blackboards that no one understands outside of the world of math. There is no way for me to assess the veracity of what is there.

    Also so much of what is discovered is changed or added to or subtracted from later on when our understanding grows. But in math is that also true? Do some things we thought added up actually don't? Happy Tuesday to thee and thine Durdle! :)
      June 8, 2021 3:00 AM MDT