Mathematics is both invented and discovered

I'm fascinated by the question of whether the laws of nature are "out there" in an objective external world, or "in here" within the subjective confines of the human brain.

A recent post on my other blog about male/female conversation styles mentions how I'd talk about this topic with another philosophically-minded man.

When men talk, most of the time they aren't trying to either reveal, or gain access to, inner feelings. My wife and I used to get another with another couple. The other guy and I would converse in one corner of our living room, while the wives huddled on the couch.

Our male conversation always focused on Grand Cosmic Subjects, like whether the laws of nature are actually "out there," or whether they're a manifestation of the human mind. We'd learn a lot about each other in this fashion.

Just not the same things the women would learn about.

After the other couple left, my wife would say something like, "How is Michael handling the death of his father?" I'd say, "He never mentioned anything about it. I didn't even know his father died."

This would astound Laurel. It seemed perfectly natural to me.

I bring up this personal vignette because it mirrors a long-standing debate among mathematicians: is their field of study invented or discovered? If invented, there's a subjective aspect to the laws of nature -- many of which, especially in physics, can be accurately modeled by mathematical equations.

This would be like my wife and her friend choosing to talk about one topic, while my friend and I choose to talk about a different topic. The content of our conversations would flow from what we wanted to focus on, not on an immutable Law of Living Room Discussions.

But many mathematicians are Platonists, believing that they are discovering numerical truths independent of the human mind. Here's how theoretical astrophysicist Mario Livio puts it in a fascinating Scientific American article, "Why Math Works."

At the core of this mystery [how math captures the natural world] lies an argument that mathematicians, physicists, philosophers, and cognitive scientists have had for centuries: Is math an invented set of tools, as Einstein believed? Or does it actually exist in some abstract realm, with humans merely discovering its truths?

Many great mathematicians -- including David Hilbert, Georg Cantor and the group known as Nicolas Bourbaki -- have shared Einstein's view, associated with a school of thought called Formalism. But other illustrious thinkers -- among them Godfrey Harold Hardy, Roger Penrose, and Kurt Godel -- have held the opposite view, Platonism.

So in a way the debate over the nature of mathematics is sort of like arguments about God. Is there really something we call "God" out there in some hidden sphere of existence, or is this notion a fabrication of the human mind?

Of course, there's a big difference between mathematics and God: the mathematical laws of nature accurately model many aspects of the natural world, while hypothesizing "God" doesn't add anything to our understanding of phenomena experienced in everyday life.

Still, I think those of us interested in the God-question can learn something from how Livio considers the issue of whether mathematics is invented or discovered. First I'll share his conclusion:

This debate about the nature of mathematics rages on today and seems to elude an answer. I believe that by asking simply whether mathematics is invented or discovered, we ignore the possibility of a more intricate answer: both invention and discovery play a crucial role.

I posit that together they account for why math works so well. Although eliminating the dichotomy between invention and discovery does not fully explain the unreasonable effectiveness of mathematics, the problem is so profound that even a partial step toward solving it is progress.

Let's be clear: as noted above I don't see any evidence that adding God or any other divinity into a description of non-human reality works at all, much less "so well." I just want to point out how even mathematics, the foundation of our most successful scientific discoveries, arguably has a subjective, invented component.

In that regard, I found this paragraph in Livio's article to be deeply thought-provoking.

Michael Atiyah, one of the greatest mathematicians of the 20th century, has presented an elegant thought experiment that reveals just how perception colors which mathematical concepts we embrace -- even ones as seemingly fundamental as numbers.

German mathematician Leopold Kronecker famously declared, "God created the natural numbers, all else is the work of man." But imagine if the intelligence in our world resided not with humankind but rather with a singular, isolated jellyfish, floating deep in the Pacific Ocean.

Everything in its experience would be continous, from the flow of the surrounding water to its fluctuating temperature and pressure. In such an environment, lacking individual objects or indeed anything discrete ,would the concept of number arise? If there were nothing to count, would numbers exist?

A few years ago I conducted my own thought experiment that went even further than reducing our world's consciousness to that of an isolated jellyfish. I said, Consider a cosmos with no consciousness.

When I pondered this, I had the same problem that popped up when I read about Kronecker's thought experiment. Mention of the jellyfish made me think, "Wow, what a stupid creature that would be. It wouldn't know what exists beyond the ocean. It wouldn't be able to perceive all the sights, sounds, smells, and such on land, in the air -- everything we humans experience."

But then I realized that my perspective was just that: mine. I was looking upon the jellyfish from my concious human point of view, even though the thought experiment was to envision all of the world's consciousness being embedded in a jellyfish.

In my blog post about a cosmos with no consciousness, I said:

Still, my thought experiment has led me somewhere, though not to my intended destination of an imagined cosmos with no consciousness. I’ve understood that different sorts of consciousnesses are aware of the cosmos in different ways. We may not create reality, but our own unique perception of it is indeed created.

This is pretty much the same conclusion that Mario Livio arrived at in considering the question of why math works.

Mathematics is an intricate fusion of inventions and discoveries. Concepts are generally invented, and even though all the correct relations among them existed before their discovery, humans still chose which ones to study.

Ditto with God, though to a much greater degree.

Notions of God, supernaturalism, spirit, soul, enlightenment, salvation, heaven, and such are all conceptual. They're invented by people. There is no evidence, in contrast with mathematics, that seekers of the divine ever come into contact with anything but mental inventions.

So if even mathematics is a fusion of subjectivity and objectivity, religions are vastly more so. In fact, religiosity may be entirely invented by humans, with not a bit of discovery thrown into the dogma recipe.

I could be wrong about this. That's the beauty of a scientific outlook on life. You're always willing to say, "I could be wrong."

Comments

[Marina]

Very interesting thread.

I never thought if maths was invented or discovered. I was good at maths when I was at school and it was one of my favourite subjects because there was really no learning to it apart from a wee bit of formulas and equations here and there. And the real beauty of it was, you didn’t have to study anything as long as you understood the principle of whatever ‘sum’ you were doing.

The first thought that comes to mind is – it has to be discovered. I mean, we seem to discover everything else; from gold, oil, our spouse has been cheating on us, the law of gravity etc.

I’d like to know what was ever discovered that wasn’t already there. Ok, we can discover we have being fooling ourselves but what really have we invented? I suppose you could say we invented this ‘fooling’ ourselves. But then, were not all these thoughts and words not just a different mix of words that we put together in another way?

Oh sure, we can say we invented the telephone, the car, all sorts of weird and wonderful gadgets.

But what have we actually invented? Where all these parts and components not already there in some form or other and with a wee bit of mixing this with that, and putting this part along with that.
Hey presto, we now have something new, or should we say recycled - a car and some petrol we have discovered to put in this new ‘invention’. We have invented a whole lot of things by discovering how to mix things together.

I am not sure about the ‘laws of nature’ either. I mean I can’t change these laws of nature to suit my subjective self. No amount of wishing for a ‘sunny’ day seems to make it happen.

I could hazard a wild guess and say the laws of nature is both subjective and objective, inside lil old me and outside lil old me. Have we not laws of nature working internally? And maybe this is a reflection in the outside objective world on a bigger scale. Then maybe it is neither or, maybe it just is. And it is like, if we ‘mess’ too much with these ‘laws’ a natural disaster may be caused. But again, ???

You could say math is an invention of energy, space, distance .... that has been already discovered, then formulated to make a new invention or new discovery, in a different way, with a different perspective.

I don’t think either way anyone can ‘win’ this argument. And I sure as hell don’t know.

Bringing God into the equation, well, again the debate on both sides is forever going on. At this stage I am leaning (not grasping) onto the ‘notion’ that it is inside to be discovered. Again, we have invented many Gods and see them as objectively ‘out there’. Like maths, who is right? I think both can be true from different perspectives. If they say we are all God, then He/She/It/Presence/Aliveness/Oneness...... can be subjective and objective both at the same time.

There is a difference between a notion of god and an experience of god. There is so much arguing and misunderstandings because we have a tendency to hold on to our own (subjective) experience and see that, that alone is true. Do that with anything – wham, separation, division. 1 becomes 2 and hence the math start!

Do you ever notice, all these paradox’s don’t seem paradoxical until you talk or thing about them?

I laughed on the woman, man relating stuff. So funny! It would make a really good film. It reminds me of the film Chancy Gardiner. I know it wasn’t about man/woman relating but then Chancy wasn’t an ordinary ‘guy’. :))

Marina

[Mike Williams]

The Great Secret of OZ

Gurinder may have a hard time in Spanish stock.
Especially since Spain will collaspe.

Spain's trillion dollar debt will sink the fiat
currencies of world, as the same banks that own
our Federal Reserve, own the European central banks.

Silver was hit in an after hours short trade of
$10 billion dollars a few months back in an attempt
to drop the price of gold. Gold did follow, but not
very much.

Silver dropped from $49 an onze to $32 an onze, but
gold only dropped $120 an onze.

JP Morgan sold to attempt to drive down
silver (and therefore gold). Silver is a thin market.

The Federal Reserve banks have manipulated silver
and gold for decades by their own admission.

Why? So, people accept fiat currencies, which are
backed by absolutely nothing.

The central banks don't take money out of their pockets
and loan it to us, they manufacture money out of thin
air and loan it to us.

If we eliminate the federal reserve banks most our
debt is gone, because they never loaned us any of their
money, just printed bogus money.

The world fiat currencies will collaspe in the next
5 years. There has never been a fiat currency that
lasted 40 years since the time of the Roman Empire.

The USA in at its 38th year, from when we went off
the gold standard, which used to back our dollars.

The only way to save America now is to abolish the Federal
reserve bank and all the bogus debt with it.

At its founding the USA printed its own money and
therefore had no debt on printed money to pay back.

http://www.youtube.com/watch?v=swkq2E8mswI

[Mike Williams]

The Great Secret of Oz (cont)

Obama' largest contributor was
Goldman Sach's, one of the owners
of our federal reserve bank.

One of the other largest contributors
was J P Morgan, another of
the owners of our federal reserve bank.

Look at the past presidents and it is
the same.

The banks dominate Washington, control
our lives and finace the wars.

That own our news networks, which keep
us ignorant.

We are completely manipulated, yet
believe we are free.

Joseph Stalin said, "It matters not
whom you vote for, you are always voting
for one of us."

[Marina]

Good post Mike.

Here is a quote from Michael Hudson from link below. Apparently biggest 'robbery' in history.

"The Obama administration raised the financial sector’s bailout to $13
trillion. This has vastly increased the government debt. And now, Mr.
Obama wants to bring it back down by cutting back Social Security,
Medicare, Medicaid and other social spending – to transfer wealth and
income to the top of the economic pyramid.

This is how he is doing what politicians are supposed to do:
delivering his constituency (liberals, racial minorities, urban
dwellers and the poor – in fact, the American mainstream) to his
campaign contributors."

http://michael-hudson.com/2011/07/the-euthanasia-of-industry/

Marina

[Mike Williams]

"If you shut up the truth and bury it under the ground, it will but grow, and gather to itself such explosive power that the day it bursts through it will blow up everything in its way."

- French author Emile Zola

http://www.youtube.com/watch?v=NO24XmP1c5E&feature=related

[Mike Williams]

Why are Indians and China buying 60% of the Gold ?

The stock market just experienced the worst week
in a year.

A couple months ago someone (federal reserse bank)
shorted after hours, an equilvalent of half the
availble silver on the earth in one minute,
trying to indirectly knock down the price of
gold which failed. Today gold is at a record
high of $1627 an oz. The federal reserve is
panicking.

The budget impasse is set for Aug. 2, a couple
days away.

The largest stock houses in the world predict
this, if the debt ceiling is not raised.

A 30% drop in stock prices and a drop of
5% in our Gross Domestic Product. A White House
insider had said they were upset the stock
market didn't drop more ! (CNN)

The House and Senate passed the time they
can get a deal passed by Aug. 2 on Friday (yesterday).

Obama would have to do a short term extension
even if they agree.

Moody's and S&P have said they are about to downgrade
USA debt below AAA.

But, since Gurinder is Indian, don't worry about
his finances. Indians love gold and have lots
of it.

He will be around to survive the crisis.
Obamas financial men are loaded with Goldman
Sach's boys including Geithner. Don't worry,
his other branches are loaded with them also.

So, with the fate of our country now in
the hands of Goldman Sachs, what could
we possibly have to worry about ?

(Goldman Sachs is one of the owners of our
Federal Reserve Bank, Goldman Sachs was Obamas largest political contributor)

[George]

It seems M-theory is being held out to the the theory of everything.

In particular, it provides explanations for some of the biggest dilemnas in physics: what caused the big bang (branes colliding), why gravity is so weak (diffusion from our to another brane) and also why the laws of physics in our particular universe are just the way they are so as to allow structure and lifeforms to exist (multiverse and anthropic fine-tuning).

However, there's not much in the way of emperical evidence and there might never be. This raises two big questions.

1. Is science ultimately limited by its very own emperical method in that some phenomena (in different universes) can never be observed?

2. Will cutting-edge science progress along explanatory grounds rather than evidentiary ones? String theory is more a combination of philosophy and abstract mathematics (albeit elegant and consist), rather than emperical evidence. If science does proceed on such grounds this would seem to give more weight to the platonic arguement of mathematics dismissed above.

Taking a giant step from the mathematically platonic to the mystical, is it coincidence that one of the key hallmarks of M-theory is the necessity for hidden dimensions and parallel universes? The mystics have spoken of hidden realms (or aspects of reality). M-theory predicts 11 dimesions, we can experience 4 (three spacial dimensions x, y, z and time) and 7 other hidden dimensions. At the risk of making a cloud-cuckooland connection that will get every mystic and his dog jumping up and down shouting 'we told you so' - aren't there 7 hidden realms or spheres taught in RS cosmology?

[George]

If you are interested, there's an hour-long documentary on the BBC IPlayer website called "Code", by a mathematician called marcus de sautay.

He seems to believe there is a code (mathematical realm) that underpins all reality and this would also support a more platonic view. Fascinating.

[Mike Williams]

Hi George,
What makes string theory so facinating
if that it used two different sets
of data to arrive at the exact same number.

Completely different variables.

But, it may be thousands of years
before we can prove it.

I believe it will turn out to be correct.

11 dimesions. Not to be confused with
the inner planes of yoga, which do not exist.

[George]

Mike,

Yes, but there were 5 ST equations in 10D, before Witten showed them to be 5 ways of looking at the same equation in 11D. The 11D solution had been already proposed by the alternative 'supergravity' theory of the 70s.

Still only a hypothesis insofar as emperical evidence is concerned, so not science yet but elegant maths. ST is yet another in a growing list of theories that is dependent on the existence of parallel universes. The multiverse has been used to explain the anthropic fine-tuning of our universe and in the QM by Everett discarded in favour of Bohr's copengahen interpretation.

Parallel universes and other realms or dimensions of existence are what mystics have always claimed. As for the existence of these worlds or planes, whether claimed by yoga or ST, neither has a shred of emperical evidence therefore.

[Mike Williams]

" As for the existence of these worlds or planes, whether claimed by yoga or ST, neither has a shred of emperical evidence therefore."
quote George

That's important for people to know.
Faquir Chand confirmed this.

I spent years of my life studying cycles.
Developed a very sophisticated computer
system on a new theory that was incredibly
accurate from micro to macro.

The Foundation for the Study of Cycles
20 years ago asked me to be the vice president.
I didn't take it.

I called my new scientific discovery
Interdimensional Cycles. Used to run full page
adds in the top major stocks and commodites magazines.

It has been developed into the most
sophisticated trading system over the
last 20 years, to the group I sold it
to.

Absolutely no question inner dimensions
exist to my mind. We are being
bounced around unconsciously by very powerful forces.

I came to the conclusion most people
are basically unconscious despite what
they may think.

[Nietzsche]

Mike
Chomsky analyzed the ignorance of the American people towards the foreign policy. He concluded that there is a mechanism that if a journalist writes honest about brutal foreign policy actions to lets say the arabs in Iran by example than he will not get promoted like the others. He is not openly forbidden to write but he is controlled by more subtle mechanisms. This explains also why most Americans where so surprised to see the 9/11 actions they honestly did not even know that they where hated by most Arabs and they wondered what they ever did to them.
If the papers would have been honest than al least there would have been some understanding that some actions are a reaction to exploiting the resources and controlling the governments in the rest of the world also called imperialism. After wo2 America has been a very big brother to the world and the people don't even know about it. Now we are turned into paranoids that have to search each and every eastern looking person because that is all we know about the enemy while the real enemy is the lack of honest publishing in the papers. Big brother is even controlling the people there.
Not impossible that there are a hand full of people behind the curtains spreading all this evil to fill their pockets even more or whatever sinister motivation.

[Nietzsche]

Another small point.
If M-theory explains what caused the big bang than what caused that thing that caused the big bang?
Isn't it time for the scientist to study the antinomy of causality like Kant did a long time ago? They are looking for the cause of causality while they don't see that it is on there nose all the time if you get this metaphor ;)

[Nietzsche]

I think that there are not even three dimensions out there. Someone cleverly found out that when he made sounds about the world around him that he could make a different sound for the depth and for the width and for the hight of an object and this person was the first mathematician that invented natural language as it is still today about object of three dimensions. Now the mathematicians studied our language and made it more formal in to mathematics and someone again got a brain wave and introduced 10 other dimensions to talk even more elegant and efficient about an object.
Now somehow it is more practical because of our limited processing power to talk in three dimensions but who knows slowly we might all talk in 10 dimensions one day as we find ways to chemical increase our processing power :)

[George]

The mystic says there is a hidden reality to our normal experience, tho its unclear whether this means hidden dimensions to reality OR to our minds, or both.

There are many powerful forces that are hidden and of which we are not consciously aware, but the mystic requires a cosmic intelligence or consciousness underlying such hidden aspects.

Human brains have evolved in local conditions (earth) of our universe to thus make sense of a limited portion of our universe. So its unsuprising we intuitively struggle to make sense of more universal laws or parallel universes, or that there exist hidden aspects of reality of which we have little understanding or control.

However, humans brains have evolved to a level of complexity that is capable of abstract thought. This allows us to view or imagine hidden aspects of reality that we have not experienced, sensed or previously aware of. It also allows us to view ourselves (self-aware) and ourselves in relation to reality.

As such, this complexity seems to afford humans at least some control (and awareness) over hidden aspects of both ojective physical reality as well as our subjective mental reality.

While there may be limits to our control over reality, it seems that as our abstract knowledge base (science and psychology) grows so does our degree of control.

The many different human cultures that exist show the flexibility of our complex brains. While other animals might be almost solely controlled by their natures, humans appear to be far more predisposed to learned behaviour that can be controlled to an extent.

The complexity of the human brain allows us to imagine hidden realms and an abstract perspective beyond our narrower experience, but the price seems to be a tendency to get lost in such imagined abstract realms that might be divorced from reality. This is surely why science is needed to anchor our imaginations and seperate theory from reality. Tho I suppose the mystical argument is that they somehow go even further and that there is a different aspect to the human makeup, in which the mind is stilled, to supposedly arrive at an even more fundamentally hidden experience.

[George]

Nietzche,

yip causality seems central to science and yet there are scientific theories which question this itself, i.e. QM uncertainty, the information paradox, etc.

M-theory supposedly answers what caused the big bang (two colliding membranes) AND it implies there was time before the bang.

M-theory supposedly overcomes two big problems of the old BB singularity theory:
1) it explains how galaxies cluster, which was impossible given the amorphous (uniform) fireball assumption. Instead, each of the two giant membranes have rippling surfaces, and thus collided at multiple points of energy release, explaining clumped galactic structure.
2) The laws of the universe broke down at the moment of the singularity. M-theory avoids the singularity and thus the laws of the universe can be wound back to the moment of the bang and before.

But you are right it doesn't explain what caused the membranes themselves. It also still leaves the problem of how something can be spontaneously created from nothing. And finally it does not explain the laws of the multiverse; it may be that each universe has a unique set of laws but what is the overarching law of the multiverse according to which each universe generates these different laws?

[Nietzsche]

George I would be careful with abstract thinking because it makes you vunerable to sects and religion too. Often the intelligent are the first victims.

Thanks for the information on M-theory that was not known when I studied physics some time ago.

[George]

Yes Nietzche, perhaps abstract thinking is responsible for both science and religion, but also maybe neoplatonic thought, imagination, poetry, transcendence, etc - and many other facets that seems to distinguish humans.

The M-theory stuff is brand-new, always was a bit circumspect of string theory, but it does seem that M-theory has a little more to it. Could turn out to be nonsense since got no emperical evidence, but the maths sounds just too elegant and it explains many problems in other emperically-supported theories to be too far off the mark.

Kant is a big part of philosophy of science, but i've struggled to understand what it is exactly he is trying to say about pure reason. any way to explain it simply?

[Mike Williams]

Stock Market Crash today caused by
bank run in Italy.

This has been blacked out in USA media
for obvious reasons.

For the last few days Europe has seen
a run on the banks. Italy is reportedly
out of cash now.

Italy has the third largest national debt
behind USA and Japan.

If Italy goes, the entire world banking
system will collaspe. Larry Kudlow of
MSNBC reported Italy could fail Friday
if the European bankers can't fix it
fast.

USA goverment must be blocking this in
the USA as not to cause panic.

I went to my bank and tried to take
out $10,000, as I do for coin shows.

They would only give me $3,000 and all
they had left was $20 bills.

The bank teller told me not to feel
bad, the next person would get theirs
$3,000 in $5 bills.

J.P. Morgan said Spain will run out of money
in February and had thought Italy would
not fail till September.

[Nietzsche]

I would like the very old gold dinar back. I hate the principle of money. Why not go to the market with gold or chicken and buy food. Some people do just that and they don't suffer the uncertainty of money that much.

About Kant. The best way to understand him is to read a little about him in books about the history of philosophy. But also get a copy of 'Critigue of judgement' and read some parts of it. It is well indexed.

My understanding is that Kant tries to distinguish between stuff that is out there and stuff that we add to it during the proces of perception. Much like looking at something with glaces. My current tv has 3d glaces. If I look at the tv without the glaces I see a lot of rubbish. If I put on the glaces I perceive three dimensional space information. Does this mean that space is in the flat tv or does this mean that space was created during the proces of perception? Obvious it is in the glaces in this case :)

Kant thought that if we can not think of the absence of something like space or time or causality than it might mean that it is not really out there. If you have a spot on your glaces than wherever you look that spot will show. But if the spot is out there than it will only show now and than.
We always see time, space and causality no matter what we try. We can't shake it away that is a clue that it might be our perception.
But also important is that it solves a lot of paradoxes to assume that space time and causality are within our perception. It is an elegant hypotheses often compared with the insight that the earth revolved around the sun instead of the sun revolving around the earth.