an ode to the pillars of mathematics.

jessica commented re: my last entry that:
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Here's a question for you: I run up against claims that mathematics isn't certain. Last week I heard a talk, "Mathematical accidents and the end of explanation." Or what about Morris Kline and "Mathematics: the loss of certainty."

I know you used to have a view of the certainty of math, in your why you like math essay from a while ago, but I figured your opinion had changed. Do math people address these issues, or are they just outside attacks? (NB: these people don't dimiss the worth of math)

I don't exactly understand what they say.
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my response:

I have never heard a talk or been made aware of this book... in searching for it, I found some reviews and comments that made more clear to me what it was talking about.

i will quote heavily from a review by a CS prof at stanford (found here: http://www-formal.stanford.edu/jmc/reviews/kline/kline.html)

quote 1:
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Professor Kline recounts a series of ``shocks'', ``disasters'' and ``shattering'' experiences leading to a ``loss of certainty'' in mathematics. However, he doesn't mean that the astronaut should mistrust the computations that tell him that firing the rocket in the prescribed direction for the prescribed number of seconds will get him to the moon.

The ancient Greeks were ``shocked'' to discover that the side and diagonal of a square could not be integer multiples of a common length. This spoiled their plan to found all mathematics on that of whole numbers. Nineteenth century mathematics was ``shattered'' by the discovery of non-Euclidean geometry (violating Euclid's axiom that there is exactly one parallel to a line through an external point), which showed that Euclidean geometry isn't based on self-evident axioms about physical space (as most people believed). Nor is it a necessary way of thinking about the world (as Kant had said).
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In this sense, the shattering of mathematics, is merely showing that the original asumptions about how things turn out to be are not always true. (e.g. finding the length of the diagonal of a square should be an integer, there existing non-Euclidean geometry (Euclidean geometry is what you learn in Geometry in high school, but there do exist geometries that don't obey the same laws you learn in school... there are theories that the universe follows a different geometry, but we just see such a small corner of it that locally it looks euclidean). I have no problem with this, and think that it doesn't show inconsistency -- it shows a developing of understanding. It's not as if someone had PROVED the diagonal of a square must have integer length if the sides have integer length. It's an idea people hoped to be true and were shocked when it was proved it wasn't. This isn't an issue of inconsistency; it's an issue of people not fully realizing the complexity of the universe we live in/the complexity of the structure of mathematics.

However, there DO exist questions of certainty in mathematics

quote 2:
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Once detached from physics, mathematics developed on the basis of the theory of sets, at first informal and then increasingly axiomatized, culminating in formalisms so well described that proofs can be checked by computer. However, Gottlob Frege's plausible axioms led to Bertrand Russell's surprising paradox of the the set of all sets that are not members of themselves. (Is it a member of itself?). L.E.J. Brouwer reacted with a doctrine that only constructive mathematical objects should be allowed (making for a picky and ugly mathematics), whereas David Hilbert proposed to prove mathematics consistent by showing that starting from the axioms and following the rules could never lead to contradiction. In 1931 Kurt Goedel showed that Hilbert's program cannot be carried out, and this was another surprise.

However, Hilbert's program and Tarski's work led to metamathematics, which studies mathematical theories as mathematical objects. This replaced many of the disputes about the foundations of mathematics by the peaceful study of the structure of the different approaches.
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Let me summarize some developments in math in the last century that are referenced above. Do you believe that sets exist? Can you make a set of objects, and another set, find their intersection and union? Sure you can. At least if you're only looking at a finite number of objects. Hopefully you learned that in grade school or high school or something. So sets, exactly what you learned about back in the day... we're on the same page.

Think about the set of all sets which are not members of themselves. Does it contain itself?

If this sets does NOT contain itself then it doesn't really contain ALL sets who don't contain themselves, does it?

If this set DOES contain itself, it's no longer a set of sets not containing themselves, so again it violates its own definition.

paradox.... in a field that's supposed to be completely based on building up truths from axioms.... oi.

thus there are issues with sets, which are a fundamental basis of mathematics.

another example: the banach-tarski paradox says that i can take a sphere, cut it up into a bunch of pieces and put them back together in such a way to get two separate spheres OF THE EXACT SAME SIZE AS THE ORIGINAL SPHERE. this isn't just wishful thinking or a fluke, this has been proven rigorously with topological arguments.

something strange is going on.

here's where you get into logic -- the part of math that my friends scott and sam and paul worry about. back around the turn of the century people saw things like the above two paradoxes and were worried about mathematics and its consistency. why doesn't something as simple as "the set of all sets" seem to coherently exist? why can i theoretically saw up my bouncy ball appropriately and get two? mathematics takes pride in taking axioms and building up theorems from them, and proving them rigorously. mathematics is not an experimental science where we run something through tests and see the same result enough times and declare it true! every step in a mathematical proof must be already proven to be true, or be a fundamental axiom/definition of the universe (e.g. 0+0=0). this created a flurry of activity 100-ish years ago. people wanted to prove mathematics extremely pedantically and try to make sure all strange things (like the two examples above) would go away. here you have books like principia mathematica coming out. the mathematicians saw their science as the epitomy of consistency and truth once you adopt the right model, but unlike the "goofs" and "inconsistencies" of quote 1, these seemed more fundamental.

1931. Kurt Godel comes on the scene. He proved that any formal system (of sufficient expressive power) cannot prove itself to be consistent. In other words: I give you a set of axioms and let you go off and start proving all the things you possibly can given those axioms. No matter what you do, THERE WILL ALWAYS EXIST TRUE BUT UNPROVABLE STATEMENTS in your system. This basically showed that the recent efforts of math world (trying to pedantically axiomatize everything) were in vain. (here you can find notes to a math class that talked about this paradigm-shattering theorem).

Here's another summary of Godel's work: http://www.chaos.org.uk/~eddy/math/Godel.html Again, they state the conclusion of Godel's work is that any formal system including Peano axioms necessarily (a) could not be both consistent and complete; and (2) could not prove itself consistent without proving itself inconsistent. You may be surprised to hear this but Peano axioms are the basic principles of doing arithmetic. (see here: http://mathworld.wolfram.com/PeanosAxioms.html for a definition -- notice that there's nothing in them that you don't already take for granted about the way your world works)...

So, at its core, Godel showed mathematics (or any system containing basic arithmetic) will never be complete. No matter the system, there are statements we cannot prove or disprove. Lots of people didn't like that idea to start. It bends many minds today, but 70 years later, people have learned to work with it.

When you start talking about the nitty gritty, mathematicians might say to you that "oh all math is fine... as long as you accept the axiom of choice... or as long as you accept zorn's lemma"... (the axiom of choice says that "Let X be a collection of non-empty sets. Then we can choose a member from each set in that collection." -- seems pretty intuitive, right?.... zorn's lemma says "If S is any nonempty partially ordered set in which every chain has an upper bound, then S has a maximal element." Basically it says if you have a set and you can give an upper bound on how big certain subsets are, you can find a maximal element... again, seems pretty intuitive) (quote from here: "In effect, when we accept the Axiom of Choice, this means we are agreeing to the convention that we shall permit ourselves to use a choice function f in proofs, as though it "exists" in some sense, even though we cannot give an explicit example of it or an explicit algorithm for it. ")... In effect, mathematics assumes such principles as "hey, I can start a proof with 'Pick a number (or other mathematical object).'" and go from there to try to take care of the above issues.

You (Jessica) originally asked about the certainty of math, and here i've gone rambling about lots of theorems and paradoxes and ideas. In summary, 90+% of mathematicians would tell you Godel proved mathematics would never be complete -- there will always exist questions that we cannot prove are true even though they are. However, those questions are dwarfed (philosophically) by the body of things we CAN prove and DO know. Any mathematician will recognize that unprovable truths exist; however, the sweeping majority of mathematicians are not troubled by this in their work. Everything that HAS been proven IS certain. It has been proven rigorously from axioms and from other theorems built up from those axioms. This isn't to say that mathematics is a clear-cut a body of knowledge as one might think; because we have our proven paradoxes and issues like above. But it does not say that the work that's been done for ages in uncertain. We just have to recognize that there do exist out there statements that are true and NEVER will be able to be proved.

Finaly paragraph from the stanford review I've refered to throughout?

quote 3:
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Professor Kline's presentation of these and other surprises as shocks that made mathematicians lose confidence in the certainty and in the future of mathematics seems overdrawn. While the consistency of even arithmetic cannot be proved, most mathematicians seem to believe (with Goedel) that mathematical truth exists and that present mathematics is true. No mathematician expects an inconsistency to be found in set theory, and our confidence in this is greater than our confidence in any part of physics.
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In conclusion, yes we recognize inherent incompleteness to any formal system such as mathematics. No, this is not a problem for us. As stated above, the sweeping majority has the view that we are still in search of truth and what we prove is true and beautiful (and I still argue that math is one of the very few if not the only field where one can study absolute truth with certainty in that the proofs we do have are either true or false, and never gray). When we say certainty of mathematics, we have to be clear. If it's that original assumptions about the world have been shattered (e.g. the diagonal of a square example above), that's not so much certainty as mis-assumption. If it's that we're talking about the inherent paradoxes of math (e.g. the set question or the banach-tarski paradox above), those are recognized and their existence is closely tied in with Godel's proof, which again is accepted clearly -- there exist quirky things in mathematics. However, this does not ruin the certainty of proofs derived from the axioms we do have.

Congrats if you made it to the end! :-) I'll stop rambling now :-)

*thanks to scott schneider for proofreading this post and making sure i didn't tell any blatant lies about set theory :-)