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Posts Tagged ‘mathematics’

Previously I made this set of statements:

Computation irreducibility, the principal (unproven), suggests the best we are going to be able to do to understand EVERYTHING is just to keep computing and observing. Everything is unfolding in front of us and it’s “ahead” of us in ways that aren’t compressible. This suggests, to me, that our best source of figuring things out is to CREATE. Let things evolve and because we created them we understand exactly what went into them and after we’re dead we will have machines we made that can also understand what went into them.

This is a rather bulky ambiguous idea without putting some details behind it. What I am suggesting is that the endless zoological approach to observing and categorizing “the natural world” isn’t going to reveal path forward on many of the lingering big questions. For instance, there’s only so far back into the Big Bang we can look. A less costly effort is what is happening at LHC, where fundamental interactions are being “created” experimentally. Or in the case of the origin of life, there’s only so much mining the clues of earth and exoplanets we can do. A likely more fruitful in our lifetime approach will be to create life – in a lab, with computers and by shipping genetic and biomass out into space. And so on.

This logic carries on in the pure abstraction layers too. Computational complexity studies is about creating ever new complex systems to then go observe the properties and behaviors. Mathematics has always been this way… we extend mathematics by “creating” all sorts of new structures, first we did this geometrically, then logically/axiomatically, and now computationally. (I could probably argue successfully that these are equivalent)

All that said, we cannot abandon observation of the world around us. We lack the universal scale to create all that is around us. And we are very far from exhausting all the knowledge that can come from observation of what exists right now. The approaches of observation and creation go hand in hand, and for the most important questions it’s required to do both to be anywhere close to certain we’re on the right path to what might actually be going on. The reality is, our ability to know is quite limited. We will always lack some level of detail. Constant revision of the observational record and the attempt to recreate or create new things we see often reveals little, but critical details we miss in our initial assessments.

Examples that come to mind are Bertrand Russell’s and Whitehead’s attempt to fully articulate all of mathematics in Principia Mathematics. Godel undid that one rather handedly with his incompleteness theorem. More dramatic examples from history include the destruction of the idea of a earth centered universe, the spacetime curvature revelations of Einstein and Minkoski, and, of course, evolutionary genetics unraveling of a whole host of long standing theories.

In all those examples there’s a dance between observation and creation. Of course it’s way too clean to maintain there’s a clear distinction between observing the natural world and creating something new. Really these are not different activities. It’s just a matter of perspective on how where we’re honing our questions. The overall logical perspective I hold is that everything is a search through the space of possibilities. “Creation” is really just a repackaging of patterns. I tend to maintain it as a different observational approach rather than lump it in because something happens to people when they think they are creating – they are more open to different possibilities. When we think we are purely observing we are more inclined to associate what we observe with previously observed phenomenon. When we “create” we’ve already primed ourselves to look for “new.”

It is a combination of the likely reality of computational irreducibility and the psychological effect of “creating” and seeing things in a new light that I so strongly suggest “creating” more if we want to ask better questions, debunk false answers and increase our knowledge.

 

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There is truth.   Truth exists.  There is a truth to this existence, this universe.   We might lack the language or the pictorial tools or the right theory and models, but there is truth.

What is this truth?  what is truth?

Things exist, we exist, there is a speed of light, the square root of two is irrational, the halting problem is undecidable, there are abstract and real relations between abstract and real things.

The truth is a something that, yes, has a correspondence to the facts.  That is not the end of it though (despite the pragmatic claims of some!).   The truth has a correspondence to the facts because it is true!   The facts HAVE to line up against a truth.   The truth exists outside of specific events or objects.   A number has an existence, if even only as an idea, and it has relations to other things.  And the description of that number and those relations ARE truth.  A computer program has its truth whether you run the program or not.  If you were to run it it would halt or not halt, that potential is in the computer program from the beginning, it doesn’t arise from it’s execution.

On Proof and not Proof but Use

We can prove these truths and many more.  We can prove through English words or through mathematical symbolism or computer programs.   Our proofs, put into these formats, can and are often wrong and set to be revised over and over until there are no holes.   No matter how fragile a proof and the act of providing proof the truth is still not diminished.  It is still there, whether we know it or not and whether we can account for it or not.  And the truth begs proof.  It begs to be known in its fullness and to be trusted as truth to build up to other truths.

BUT!

Proof isn’t always possible – in fact we’ve learned from issues in computability and incompleteness – that complete provability of all truth is impossible.   This beautiful truth itself further ensures that the truth will always beckon us and will never be extinguished through an endless assault.  There is always more to learn.

The unprovable truths we can still know and use.  We can use them without knowing they are true.  We do this all the time, all day long.   How many of us know the truth of how physics works? or how are computers do what they do?   and does that prevent their use – the implementation of that truth towards more truth?

Why?

Why defend truth?  Why publish an essay exalting truth and championing the search for truth? Does the truth need such a defense?

Being creatures with intelligence – that is, senses and a nervous system capable of advanced pattern recognition – our ultimate survival depends on figuring out what’s true and what isn’t.   If too many vessels (people!) for the gene code chase falsehoods the gene code isn’t likely to survive too many generations.   Life, and existence itself, depends on the conflict between entropy and shape, chaos and order, stillness and motion, signal and noise.  The truth is the abstract idea that arises from this conflict and life is the real, tangible thing born from that truth.  We learn truths – which processing of this thing into that thing that keep us alive, we live to learn these things. In a completely entropic existence there is nothing.   Without motion there is nothing.   In total chaos there is nothing.   It is the slightest change towards shape, order and signal that we find the seeds of truth and the whole truth itself.  The shaping of entropy is the truth.   Life is embodiment of truth forming.

So I can’t avoid defending the truth.  I’m defending life.  My life.  In defending it, I’m living it.  And you, in whatever ways you live, are defending the truth and your relation to other things.  If I’m alive I must seek and promote truth.   While death isn’t false, chasing falsehood leads to death or rather non existence.   Could there ever be truth to a statement like “I live falsely” or “I sought the false.”   There’s nothing to seek.  Falsehood is easy, it’s everywhere.  It’s everything that isn’t the truth.  To seek it is to exert no effort (to never grow) and to never gain – falsity has no value.  Living means growing, growing requires effort, only the truth, learning of the truth demands effort.

How do we best express and ask about truth?

There’s a great deal of literature on the unreasonable effectiveness of mathematics to describe the world.  There’s also a great deal of literature, and growing by the day, suggesting that mathematics isn’t the language of the way the universe works.   Both views I find to be rather limited.   Mathematics and doing math is about certain rigor in describing things and their relations.   It’s about forming and reforming ways to observe and question ideas, objectives, motion, features…. It’s about drawing a complete picture and all the reasons it should and shouldn’t be so.   Being this way, this wonderful thing we call mathematics, there is no way mathematics couldn’t be effective at truth expression.   Ok, for those that want to nit pick, I put “computation” in with mathematics.  Describing (writing) computer programs and talking about their features and functions and observing their behavior is doing math, it is mathematics.

Art has very similar qualities.   Art doesn’t reduce beyond what should be reduced.   It is the thing itself.  It asks questions by shifting perspectives and patterns.  It produces struggle.  Math and art are extremely hard to separate when done to their fullest.  Both completely ask the question and refuse to leave it at that.   Both have aspects of immediate impression but also have a very subtle slow reveal.  Both require both the artist and the audience, the mathematician and the student – there is a tangible, necessary part of the truth that comes directly from the interaction between the parties, not simply the artifacts or results themselves.

Other ways of expressing and thinking are valuable and interesting.  That is, biology and sociology and political science, and so on….. these are all extremely practical implementations or executions of sub aspects of the truth and truth expression.  They are NOT the most fundamental nor the most fruitful overall.   Practiced poorly and they lead to falsehoods or at best mild distractions from the truth.  Practiced well and they very much improve the mathematics and art we do.

What does any of this get us?  What value is there in this essay?

This I cannot claim anything more about than what I have above.   For example, I don’t know how to specifically tell someone that the truth of square root of 2 is irrational has x,y,z value to them.  It certainly led to a fruitful exploration and exposition of a great deal of logic and mathematical thinking that led to computation and and and.   But that doesn’t even come close to explaining value or what talking about its value today, in this essay, matters.

My only claim would be that truth matters and if there is any truth in this essay then this essay matters.  How that matter comes to fruition I don’t know.   That it comes to any more fruition than my pounding out this essay after synthesizing many a conversation and many books on the subject and writing some computer programs and doing math is probably just a very nice consequence.

The truth’s purpose is itself, that it is true.

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I really like this post on Good Math, Bad Math.

Beyond being mildly humorous in that cranky math person non-funny kinda way, it touches on lots of my favorite subjects: enumeration, Cantor, classic proofs, cranky math people.

The catch – and it’s a huge catch – is that the tree defines a representation, not an enumeration or mapping. As a representation, taken to infinity, it includes every possible real number. But that doesn’t mean that there’s a one-to-one correspondence between the natural numbers and the real numbers. There’s no one-to-one correspondence between the natural numbers and the nodes of this infinite tree. It doesn’t escape Cantor’s diagonalization. It just replaces “real number” with “node of this infinite tree”. The infinite tree contains uncountably many values – there’s a one-to-one correspondence between nodes of the infi To see the distinction, let’s look at it as an enumeration. In an enumeration of a set, there will be some finite point in time at which any member of the set will be emitted by the enumeration. So when will you get to 1/3rd, which has no finite representation as a base-10 decimal? When will you get to π?

FUN!

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I do wonder why the speed of light is 299 792 458 m/s instead of 400 000 000 m/s or 50 billion m/s.  I know it’s constant and like other constants the universe just sort of has them and299 792 458 m / s whether they are this value or that value, the point is, they have a constant value.299 792 458 m / s.

It’s still fun to think about.

Here are two decent resources explaining the situation with the finiteness and constance of the speed of light.

Why is the speed of light constant

Many novel ideas are found on the Internet. One not so novel notion is that Einstein was wrong and that the “lightspeed limit” is really just some international conspiracy of conservative “establishment” scientists. Those who make this point neglect the fact, however, that the deduction about the speed of light is not a result of some exotic assumptions or blind speculation, but a fairly simple consequence of some fundamental assumptions about nature: in other words, if you wish to prove that Einstein was wrong, you have to show that either elementary logic is incorrect, or that some of our basic assumptions about nature are outright false.

Here is why.

Why isn’t the speed of light infinite

The fact that space and time must get mixed up to keep the speed of light constant implies that, in some sense, space and time must be the same, despite our habit of measuring space in meters and time in seconds. But if time and space are similar to the extent that they can be converted one into the other, then one needs some quantity to convert the units–namely, something measured in meters per second that can be used to multiply seconds of time to get meters of space. That something, the universal conversion factor, is the speed of light. The reason that it is limited is simply the fact that a finite amount of space is equivalent to a finite amount of time.

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Here’s an amazing video on YouTube (this will be old news to many people, as the video got popular last year at this time).  I got the back story on this bad boy from American Mathematical Society monthly mag, Notices.  You can get more detail on the video and the creators at IMA.

It’s a video of Moebius Transformations (produced by POV-ray and, of course, Mathematica)

Now that you’ve seen the visual you can appreciate the power of visualizing data and math.  Take a look at the mathematics.  I’d say a picture is worth AT LEAST a thousand words in this case.

For those wondering why we care about Moebius transformations…

In physics, the identity component of the Lorentz group acts on the celestial sphere the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of twistor theory.

Oh, and they are COOL!

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A couple of posts ago I asked if real numbers exist (like pi).

It really doesn’t matter. and I’ve come back around (one of these mental oscillations…) to the conclusion brought to my attention on the NKS blog.

Here is the key statement:

Mathematics is a symbolic language — you can argue that none of its elements “exist” in physical reality, yet they can be used to communicate information about things which are real.

I found another statement to this effect in the classic “What is Mathematics?” by Courant & Robbins, revised by Stewart.

Through the ages mathematicians have considered their objects such as numbers, points, etc., as substantial things in themselves.  Since these entities had always defied attempts at an adequate description, it slowly dawned on the mathematicians of the the nineteenth century that the question of the meaning of these objects as substantial things does not make sense within mathematics, if at all.  The only relevant assertions concerning them do no refer to substantial reality; they state only the interrelations between mathematically “undefined objects” and the rules governing operations with them.”  What points, lines, numbers “actually” are cannot and need not be discussed in mathematical science.  What matters and what corresponds to “verifiable” fact is structure and relationship, that two points determine a line, that numbers combine according to certain rules to form other numbers, etc.

I often forget that the abstraction is not the thing.  The metaphor is not the thing.  The symbol is not the thing. Mathematics never makes assertions that it is the thing.  It is an abstraction – a description of relationships devoid of many of specific objects’ and environments’ properties.  This abstraction (and simplification) is required to make progress.  If mathematicians were to create theory that was specific to every situation, object, and environment, the world would run out of shelf space for storing all the math books and we’d gain nothing over flat out recording keeping.  In a sense, mathematical abstraction is a wonderfully useful compression of information.  The application of mathematics to a specific situation is the decompression of the abstraction.

This abstraction is so useful because it lets us focus on key relations and make progress on understanding despite our lack of complete knowledge of specific objects, environments, and situations.

This abstraction is also dangerous and/or limiting.  Not all situations in the universe are able to be described by a simplified mathematical theory.  In fact, a surprising number of very simple phenomena (theoretical, biological, physical, financial, etc.) are not mathematically compressible.  That is, a purely mathematical theory will not be sufficient for understanding in many situations.

What a relief!

Some mathematicians already experienced this relief from needing to describe the universe in some ultimate truth.   That’s far too big for any discipline to bear.

Not all mathematicians and a majority of economists, business people, vcs, media, and lay people do not recognize this limitation of mathematical theory (heck, and many other theories!).  In the US (perhaps elsewhere), business models (pro formas), stock indexes, indicators, projections, forecasts, formulas dominate our thinking on very complex phenomena.   We’ve explored this issue many times on this blog.

Understanding the universe we experience requires a combination of theories.  Math can sometimes point the way and get us going, keep us focused, or help us communicate.

Whether the real numbers exist doesn’t really matter.  The real numbers are useful for moving us forward on some problems in the real world.  Pi, as a compression of a really long number and challenging concept in geometric forms, is useful in helping us make wheels, explore space, and so much more.  That is what makes math great, even if it isn’t objective, ultimate truth.

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