Author Topic: Mathematics: is it discovered or invented? TV Documentary  (Read 269 times)

Stranger

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Mathematics: is it discovered or invented? TV Documentary
« on: October 19, 2018, 01:54:57 PM »
For those interested...

So far not a bad popularisation of the question. Liked the attempt at a popular explanation of why the infinity of the continuum is bigger than the infinity of the natural numbers (about 45 minutes into episode 2), even if it was a little simplistic.

Magic Numbers: Hannah Fry's Mysterious World of Maths

Two episodes on the iPlayer and one to come on BBC4 next Wednesday.
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Littleroses

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #1 on: October 19, 2018, 02:42:06 PM »
Maths has always been a great mystery to me. ::)
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Sebastian Toe

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #2 on: October 19, 2018, 02:52:07 PM »
Maths has always been a great mystery to me. ::)
I love maths.
I also love ornothology.
When I do both together I like to call that Owlgebra.

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Littleroses

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #3 on: October 19, 2018, 05:04:36 PM »
I love maths.
I also love ornothology.
When I do both together I like to call that Owlgebra.

I'll get my coat.....

Oh dear! ;D

A six month old baby couldn't be any worse at Maths than I am . I can just about manage 1+1, but require a calculator for anything else.
« Last Edit: October 19, 2018, 06:41:22 PM by Littleroses »
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Stranger

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #4 on: October 19, 2018, 05:42:26 PM »
A six month old baby couldn't be any worse at Maths than I am . I can just about managed 1+1, but require a calculator for anything else.

Me too. In fact, in my experience, people who are good at mathematics are rarely good at mental arithmetic. Mathematics isn't really about doing sums.
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jeremyp

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #5 on: October 19, 2018, 07:55:24 PM »
Me too. In fact, in my experience, people who are good at mathematics are rarely good at mental arithmetic. Mathematics isn't really about doing sums.
When I was about to go off to University to do my maths degree, the mother of one of my friends asked "what's involved in a maths degree then, is it doing big complicated sums?"

I thought for a minute and then said "yes."

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jeremyp

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #6 on: October 19, 2018, 08:17:50 PM »
Oh dear! ;D

A six month old baby couldn't be any worse at Maths than I am . I can just about manage 1+1, but require a calculator for anything else.
What you do with a calculator is called arithmetic. There's nothing wrong with using a calculator for doing that. Mathematics is more than that.

Take Pythagorus' theorem, for example: in a triangle with a right angle, the square of the length of the longest side is the same as the  sum of the squares of the lengths of the two shorter sides. Knowing that, if you have a right angled triangle with short sides of length 3cm and 4cm, the other side  will have a length of 5cm

 3 x 3 + 4 x 4 = 9 + 16 = 25 = 5 x 5.

That is arithmetic. Proving that it always works is mathematics. Then, understanding that your proof relied on the assumption that the surface on which the triangle was drawn is flat is more mathematics. Then exploring the properties of triangles drawn on surfaces that aren't flat is more mathematics.

Mathematicians don't need calculators much because they don't really do arithmetic. Even when they do arithmetic, they tend to avoid doing the bits that really need a calculator. For example, if you have a right angled triangle where the short sides are 1 and 2 cm, you might say the length of the long side is 2.236 and a bit but a mathematician would leave it at √5. Technically they would be more correct too.
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ad_orientem

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #7 on: October 19, 2018, 09:22:31 PM »
Maths has always been a great mystery to me. ::)

Same here.
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Steve H

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #8 on: October 19, 2018, 10:41:44 PM »
Not my forte at school, but it is fascinating - like how two infinities can be different sizes.
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Stranger

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #9 on: October 21, 2018, 10:01:50 AM »
Not my forte at school, but it is fascinating - like how two infinities can be different sizes.

There are actually an infinite number of different sizes of infinity but the most 'obvious' difference is the one given in the programme between 'countable' infinity and the continuum. The full proof actually isn't all that hard (and involves no sums) but probably too long for a popular TV programme. The problem with the explanation she gave was that you could conclude that it applied to the rational numbers (fractions) as well, which it doesn't - they are countably infinite.
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SusanDoris

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #10 on: October 21, 2018, 11:18:29 AM »
I think it is really sad that so many people are so negative  about maths, whether it is arithmetic or tackling problems. It is usually adults who reinforce any negative comment by children instead of finding the words to give them the confidence to realise what they do know and to try a bit more and realise that they can do better than they thought they could.
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Steve H

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #11 on: October 21, 2018, 11:27:20 AM »
There are actually an infinite number of different sizes of infinity but the most 'obvious' difference is the one given in the programme between 'countable' infinity and the continuum. The full proof actually isn't all that hard (and involves no sums) but probably too long for a popular TV programme. The problem with the explanation she gave was that you could conclude that it applied to the rational numbers (fractions) as well, which it doesn't - they are countably infinite.
I cottoned on to this when I was at primary school, when a teacher pointed out that there are an infinite number of radiuses to a given circle, and an infinite number of diameters, but only half as many diameters as radiuses. A better example, perhps, is positive whole numbers and positive whole even numbers. I think the reason infinity doesn't behave itself is that, while the concept is useful to mathematicians, there's no such thing: even time and space are nowadays thought to be finite, and numbers don't exist either.
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Stranger

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #12 on: October 21, 2018, 11:57:04 AM »
A better example, perhps, is positive whole numbers and positive whole even numbers.

Except that they are both countably infinite and hence exactly the same size of infinity (as mentioned in the TV programme). You can tell this because you can make a one-to-one map from one to the other:

1 <-> 2
2 <-> 4
3 <-> 6
...

You'll never run out of either, so they have the same cardinality.

In contrast, it's quite easy to prove that the real numbers do not have the same cardinality (size) because there is no such one-to-one map, even from just the real numbers from 0 to 1.

I think the reason infinity doesn't behave itself is that...

But it does behave itself - it's just not the same as finite numbers. There are lots of types of numbers and other mathematical entities that follow different rules from regular numbers,

...while the concept is useful to mathematicians, there's no such thing: even time and space are nowadays thought to be finite...

AFAIK that's still an open question, as is whether they are quantised. If they aren't quantised then you get a continuum infinity between each point.

...and numbers don't exist either.

Another open question and the main subject of the documentary that this thread is (was originally) about.
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Littleroses

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #13 on: October 21, 2018, 12:05:30 PM »
Before his brain haemorrhage my husband was always in charge of our financial affairs being highly intelligent and trillions of light years better at maths than little me. However, it is now my task to keep an eye on all our bank accounts to see nothing is amiss. I have got obsessional about it, and check them all once a day, Christmas Day included. ::)
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jeremyp

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #14 on: October 21, 2018, 12:21:29 PM »
I cottoned on to this when I was at primary school, when a teacher pointed out that there are an infinite number of radiuses to a given circle, and an infinite number of diameters, but only half as many diameters as radiuses.
Your teacher was wrong. A circle has exactly the same number of radiuses and diameters and it is the same as the number of real numbers.

Quote
A better example, perhps, is positive whole numbers and positive whole even numbers. I think the reason infinity doesn't behave itself is that, while the concept is useful to mathematicians, there's no such thing: even time and space are nowadays thought to be finite, and numbers don't exist either.

Again there is the same number of even numbers as whole numbers but the explanation is slightly easier to deal with than the circle diameter/radius case.

Mathematicians say two sets are the same size if their elements can be put into a one to one correspondence without leaving some elements from one set left out. So for the two sets { apple, tram, snake } and { Einstein, Kekulé, Newton } we can create a 1:1 correspondence (several actually). This means we can find a way to remove pairs with one item in the pair from each set and have nothing left over in either set at the end:

    apple <-> Newton
    tram  <-> Eindstein
    snake <-> Kekulé


so we know those two sets are the same size. But if the second set was { Einstein, Kekulé, Newton, Darwin } no matter what 1:1 correspondence we choose. There will always be something left over from the second set.

With infinite sets of numbers, we describe the correspondence by a mathematical relationship rather than listing all the pairs out which would take literally for ever. So, with the set of whole numbers and the set of even numbers we can pair each whole number with its double in the even numbers. This is a 1:1 relationship and by definition it mans the two sets are the same size.

A similar trick works for radiuses and diameters. Each radius crosses the circle at one point on its circumference. Each diameter crosses the circle at two points, one in the left half of the circle and one in the right half (the straight up and down diameter is a special case, we'll arbitrarily put the point at the top in the right half and the point at the bottom in the left half). So the number of radiuses is the same as the number of points on the circumference. The number of diameters is the same as the number of points in the right half of the circumference. So there are half as many diameters right? Wrong.

Imagine unwrapping the circle so that the circumference becomes a straight line. Imagine that the circumference was two units long. Now we have a line that is of length 2. All the points that were in the right half of the circle now lie between 0 and 1. All the points that were in the left half of the circle lie between 1 and 2 (technically speaking the line goes right up to but doesn't include the number 2, but this is a detail for mathematicians). So the number of diameters is the same as the number of points between 0 and 1 (not including 1 itself, see my comment about the up and down case) and the number of radiuses is the same as the number of points between 0 and 2 (not including 2 itself). Can they be put into a 1:1 correspondence. Yes they can. Just double the diameter point to get its equivalent radius point, or halve the radius point to get its corresponding diameter point.

For my next trick, I'll explain Cantor's diagonalisation proof that there are more real numbers than whole numbers, but not just now.
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jeremyp

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #15 on: October 21, 2018, 12:45:34 PM »
I think it is really sad that so many people are so negative  about maths, whether it is arithmetic or tackling problems. It is usually adults who reinforce any negative comment by children instead of finding the words to give them the confidence to realise what they do know and to try a bit more and realise that they can do better than they thought they could.

This is something that really annoys me. There's quite a stigma attached to not being able to read, so much so that people with dyslexia (I've always thought it ironic that "dyslexia" is a hard word to spell) were frequently treated as merely being stupid instead of being given the help and support needed. However, when it comes to maths, people don't think of it as an issue. There have already been three posts on this thread by people telling us they can't do maths. Sometimes (not here) people proclaim their inability at maths as though it were somehow a virtue. Maths is something nerds do. Reading is something existentialist philosophers do outside a café in Paris with a café noir and a Gauloise.

My sister-in-law (professional statistician) once ripped a new one for her son's teacher when she gave him maths to do for some infraction of the school rules. Can we please stop teaching children that maths is punishment.
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Littleroses

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #16 on: October 21, 2018, 12:53:37 PM »
This is something that really annoys me. There's quite a stigma attached to not being able to read, so much so that people with dyslexia (I've always thought it ironic that "dyslexia" is a hard word to spell) were frequently treated as merely being stupid instead of being given the help and support needed. However, when it comes to maths, people don't think of it as an issue. There have already been three posts on this thread by people telling us they can't do maths. Sometimes (not here) people proclaim their inability at maths as though it were somehow a virtue. Maths is something nerds do. Reading is something existentialist philosophers do outside a café in Paris with a café noir and a Gauloise.

My sister-in-law (professional statistician) once ripped a new one for her son's teacher when she gave him maths to do for some infraction of the school rules. Can we please stop teaching children that maths is punishment.

Our middle daughter has dyslexia, which was only discovered when she was at university. We didn't think she was quite as intelligent as her two sisters, when in fact she has a Mensa level intelligence. :-[ Her younger son has atypical dyslexia, which affects his maths as well as his reading ability. However, home schooling and home tutors appear to be helping him overcome it.
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SusanDoris

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #17 on: October 21, 2018, 01:19:10 PM »
This is something that really annoys me. There's quite a stigma attached to not being able to read, so much so that people with dyslexia (I've always thought it ironic that "dyslexia" is a hard word to spell) were frequently treated as merely being stupid instead of being given the help and support needed. However, when it comes to maths, people don't think of it as an issue. There have already been three posts on this thread by people telling us they can't do maths. Sometimes (not here) people proclaim their inability at maths as though it were somehow a virtue. Maths is something nerds do. Reading is something existentialist philosophers do outside a café in Paris with a café noir and a Gauloise.

My sister-in-law (professional statistician) once ripped a new one for her son's teacher when she gave him maths to do for some infraction of the school rules. Can we please stop teaching children that maths is punishment.
Most definitely agree. I can think of several pupils I taught whose confidence in maths I was able to establish, or re-establish.
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Sassy

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #18 on: October 28, 2018, 07:14:04 AM »
Me too. In fact, in my experience, people who are good at mathematics are rarely good at mental arithmetic. Mathematics isn't really about doing sums.

My brother-in-law was a lecturer in Mathematics at University. He was brilliant at maths but useless at English.
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Dicky Underpants

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #19 on: November 01, 2018, 04:38:23 PM »
Me too. In fact, in my experience, people who are good at mathematics are rarely good at mental arithmetic. Mathematics isn't really about doing sums.

I noticed this when I volunteered for Oxfam. Once a maths and physics teacher was entrusted with doing the cashing up - a right pig's ear he made of it.

Phyllis Tyne

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Re: Mathematics: is it discovered or invented? TV Documentary
« Reply #20 on: November 15, 2018, 02:23:02 PM »
I love maths.
I also love ornothology.
When I do both together I like to call that Owlgebra.

I think you'll find they are non-OWLverlapping magisteria.