0:00:12.486,0:00:15.730
Would mathematics exist if people didn't?

0:00:15.730,0:00:19.057
Since ancient times, [br]mankind has hotly debated

0:00:19.057,0:00:22.712
whether mathematics [br]was discovered or invented.

0:00:22.712,0:00:27.374
Did we create mathematical concepts to[br]help us understand the universe around us,

0:00:27.374,0:00:31.521
or is math the native language of[br]the universe itself,

0:00:31.521,0:00:34.734
existing whether we find [br]its truths or not?

0:00:34.734,0:00:38.102
Are numbers, polygons [br]and equations truly real,

0:00:38.102,0:00:42.676
or merely ethereal representations[br]of some theoretical ideal?

0:00:42.676,0:00:46.235
The independent reality of math has[br]some ancient advocates.

0:00:46.235,0:00:49.796
The Pythagoreans of 5th Century Greece[br]believed numbers were both

0:00:49.796,0:00:53.261
living entities and universal principles.

0:00:53.261,0:00:57.568
They called the number one, "the monad,"[br]the generator of all other numbers

0:00:57.568,0:00:59.829
and source of all creation.

0:00:59.829,0:01:02.644
Numbers were active agents in nature.

0:01:02.644,0:01:05.499
Plato argued mathematical [br]concepts were concrete

0:01:05.499,0:01:10.444
and as real as the universe itself,[br]regardless of our knowledge of them.

0:01:10.444,0:01:13.897
Euclid, the father of geometry, believed[br]nature itself

0:01:13.897,0:01:17.702
was the physical manifestation[br]of mathematical laws.

0:01:17.702,0:01:21.926
Others argue that while numbers may[br]or may not exist physically,

0:01:21.926,0:01:25.047
mathematical statements definitely don't.

0:01:25.047,0:01:29.586
Their truth values are based on rules[br]that humans created.

0:01:29.586,0:01:32.613
Mathematics is thus an invented[br]logic exercise,

0:01:32.613,0:01:36.356
with no existence outside mankind's[br]conscious thought,

0:01:36.356,0:01:40.997
a language of abstract relationships[br]based on patterns discerned by brains,

0:01:40.997,0:01:46.694
built to use those patterns to invent[br]useful but artificial order from chaos.

0:01:46.694,0:01:50.373
One proponent of this sort of idea[br]was Leopold Kronecker,

0:01:50.373,0:01:53.997
a professor of mathematics in [br]19th century Germany.

0:01:53.997,0:01:56.451
His belief is summed up in [br]his famous statement:

0:01:56.451,0:02:00.960
"God created the natural numbers,[br]all else is the work of man."

0:02:00.960,0:02:03.533
During mathematician[br]David Hilbert's lifetime,

0:02:03.533,0:02:07.131
there was a push to establish mathematics[br]as a logical construct.

0:02:07.131,0:02:10.501
Hilbert attempted to axiomatize all[br]of mathematics,

0:02:10.501,0:02:12.969
as Euclid had done with geometry.

0:02:12.969,0:02:17.525
He and others who attempted this saw[br]mathematics as a deeply philosophical game

0:02:17.525,0:02:19.700
but a game nonetheless.

0:02:19.700,0:02:23.231
Henri Poincaré, one of the father's of[br]non-Euclidean geometry,

0:02:23.231,0:02:26.238
believed that the existence of [br]non-Euclidean geometry,

0:02:26.238,0:02:30.535
dealing with the non-flat surfaces of [br]hyperbolic and elliptical curvatures,

0:02:30.535,0:02:35.001
proved that Euclidean geometry, the[br]long standing geometry of flat surfaces,

0:02:35.001,0:02:37.363
was not a universal truth,

0:02:37.363,0:02:42.051
but rather one outcome of using one[br]particular set of game rules.

0:02:42.051,0:02:45.865
But in 1960, Nobel Physics laureate[br]Eugene Wigner

0:02:45.865,0:02:50.173
coined the phrase, "the unreasonable [br]effectiveness of mathematics,"

0:02:50.173,0:02:53.283
pushing strongly for the idea that[br]mathematics is real

0:02:53.283,0:02:55.482
and discovered by people.

0:02:55.482,0:02:58.388
Wigner pointed out that many purely[br]mathematical theories

0:02:58.388,0:03:03.379
developed in a vacuum, often with no view[br]towards describing any physical phenomena,

0:03:03.379,0:03:05.873
have proven decades [br]or even centuries later,

0:03:05.873,0:03:08.337
to be the framework necessary to explain

0:03:08.337,0:03:11.440
how the universe[br]has been working all along.

0:03:11.440,0:03:15.688
For instance, the number theory of British[br]mathematician Gottfried Hardy,

0:03:15.688,0:03:19.377
who had boasted that none of his work[br]would ever be found useful

0:03:19.377,0:03:21.918
in describing any phenomena[br]in the real world,

0:03:21.918,0:03:24.660
helped establish cryptography.

0:03:24.660,0:03:26.938
Another piece of his purely[br]theoretical work

0:03:26.938,0:03:30.095
became known as the Hardy-Weinberg[br]law in genetics,

0:03:30.095,0:03:31.834
and won a Nobel prize.

0:03:31.834,0:03:34.426
And Fibonacci stumbled [br]upon his famous sequence

0:03:34.426,0:03:38.040
while looking at the growth of an [br]idealized rabbit population.

0:03:38.040,0:03:41.548
Mankind later found the sequence[br]everywhere in nature,

0:03:41.548,0:03:44.036
from sunflower seeds[br]and flower petal arrangements,

0:03:44.036,0:03:45.857
to the structure of a pineapple,

0:03:45.857,0:03:48.497
even the branching of bronchi[br]in the lungs.

0:03:48.497,0:03:52.704
Or there's the non-Euclidean work of[br]Bernhard Riemann in the 1850s,

0:03:52.704,0:03:57.291
which Einstein used in the model for[br]general relativity a century later.

0:03:57.291,0:03:58.707
Here's an even bigger jump:

0:03:58.707,0:04:02.933
mathematical knot theory, first developed[br]around 1771

0:04:02.933,0:04:05.185
to describe the geometry of position,

0:04:05.185,0:04:10.033
was used in the late 20th century[br]to explain how DNA unravels itself

0:04:10.033,0:04:12.212
during the replication process.

0:04:12.212,0:04:16.161
It may even provide key explanations[br]for string theory.

0:04:16.161,0:04:18.791
Some of the most influential [br]mathematicians and scientists

0:04:18.791,0:04:22.472
of all of human history[br]have chimed in on the issue as well,

0:04:22.472,0:04:24.093
often in surprising ways.

0:04:24.093,0:04:26.904
So, is mathematics an [br]invention or a discovery?

0:04:26.904,0:04:29.851
Artificial construct or[br]universal truth?

0:04:29.851,0:04:34.017
Human product or[br]natural, possibly divine, creation?

0:04:34.017,0:04:38.458
These questions are so deep the debate[br]often becomes spiritual in nature.

0:04:38.458,0:04:41.550
The answer might depend on the specific[br]concept being looked at,

0:04:41.550,0:04:45.177
but it can all feel like a[br]distorted zen koan.

0:04:45.177,0:04:48.806
If there's a number of trees in a forest,[br]but no one's there to count them,

0:04:48.806,0:04:50.726
does that number exist?