Math Made Almost Bearable: Garfield’s Proof of the Pythagorean Theorem part 2
June 30th, 2009
Math Made Almost Bearable: Garfield’s Proof of the Pythagorean Theorem part 1
June 10th, 2009
In this episode, Frank discusses the first half of president Garfield’s proof the of the pythagorean Theorem.
Math Made Almost Bearable: Math as a Performance Art
May 1st, 2009
In this episode I sit down with Frank Kelly to discuss what it means for math to be considered as an art, namely a performing art. Don’t worry though! on the next episode we’ll be right back to PERFORMING math for you as in episode one!
Math Made Almost Bearable: Fractions and Repeating Decimals
April 23rd, 2009
This is the first in a series called “Math Made Almost Bearable” One of my favorite professors, Dr. Frank Kelly and I are collaborating on this little project. The goal is simply to present short, interesting and intriguing facts about math in an approachable and engaging way. This first episode is called “Fractions and Repeating Decimals”.
Where is God?
April 10th, 2009
Where is God?
In the wind that laces through my fingers
And whispers in the trees -
Their furtive murmurings that speak
Of subtle mysteries.
Where is God?
With the sun that gently warms my skin
And floods the earth with light -
And by His word, relents its heat
For coolness of the night.
Where is God?
In the harmonies of well-tuned strings
And hymnals of the birds -
The choirs of creation praise
With songs that have no words.
Where is God?
Within the fibers of my heart
From whence He urges me -
‘Tis better to remember Him
Than ’tis for me to breathe.
A Garrulous Attempt at the Ineffable
April 9th, 2009
A Garrulous Attempt at the Ineffable:
The Valley of Achor
He stood on the dry, earthen edge
Silent upon a peak
Overlooking a world no longer adorned with mystery,
With ideals or land to be discovered or people to be known;
He stood, wearied, shaken, battered, buffeted, silenced,
And said, “You know not where you’ve gone
And how.
You trust nothing
And you cannot have faith in what you do not trust
And will not trust in what you will not to know
And will not to know what you cannot understand—
What catastrophes you have done, oh man,
In the name of a god you claim to understand:
Formulated, predicted, controlled,
Etherized,
Spread loosely across the impaling grass—
You’ve made a picnic out of the evening’s sky
Stole it from the children who now sleep
Under wires and vents and tubing
Wrapped in aluminum foil and red and blue lies.
Woe to me, for I am a man undone,
They have partaken of the fruit of knowledge
They mark upon themselves with variables
Greek symbols
Alphas and omegas
–undefined—
They have acted as their own gods
And turned the stone to bread—
But still cast it at the sinner,
Petrified into believing that to survive is life.
And still I sit
Like an Idiot by the murdering and the murdered
Staring out the window
And at the crossbeams that land upon a painting
In the center room—
And still I sit
Like an Idiot,
And if I were pure and honest
I fear
I might be
Tattered and irrelevant.
The fools are the geniuses,
The geniuses the fools;
They will reverse and begin again
As the centuries revolve, wash, and hang to dry again.
The words abound and proliferate
And are cast in meaning
As a letter too long for the price of postage.
The stairways spiral in a Mobius fashion
And the old man stands wishing he could start at the bottom
Of this modern day Babel
Of paved infinite regress,
And says,
‘If this is the world,
Then I disagree with the premise;
If there is a God, have mercy,
But I just disagree.’
And I
I on this edge of dirt and clay and sullied flesh
Breathing coastal wind
Believe, I think, enough
To fall silent—
Enough to be helped with my unbelief.”
~Hosea 2:15
Are We Yet Beautiful?
April 8th, 2009
“Are We Yet Beautiful?”
… … …
“God must be a painter;
Why else would there be so many colors?”
“So you’re a painter…”
… … …
Distracted from Distraction by Distraction,
I cast a glance
Toward a half-waned moon
And wheeze in a slice
Of icicled air.
These moments pierce our frigid catacomb-minds-
Cobwebbed and destitute
Untrilled, unperiled pulse;
Sterile, hollow remnants
Of conscience
Of soul
Of spirit
Of time before the end of time-
These moments of eternally resounding,
Or, perhaps, linearly, infinitely,
Uniquely sounding silence
Separate entirely from people and noise
Which move in mendacious vanity:
Thoughts of lies and conceptions of who
God ought to be based upon
Us, you, me, we
His imagery
His likeness
Those illicit discussions with dimmed eyes and loose tongues,
“And what are you doing here, my sweet image of Grace, with your careful fairness, your meek, tight, form of persuasion? What hell this must all be for you; this ball-and-chain of sunshine and rain and creation and sunder and toil and lust and love and plunder. What cruelty of God it is that He sends you to such a loose, rank, foul place as these sperm and smoke-stained walls of human mind. But please, if you must join me, suffer for me, cry for me, peel away in agony for my condition and some, some will call this salvation, some will call this love…”
We mean nothing we say.
The soul is given life by the spirit,
And the spirit by Spirit eternal;
But the tongue and brain alone
Are flesh and fleshed reduced
When given life by flesh.
-Drag in another cold breath-
We are but shadows scorned in bathos
Or life incarnate
When given level by God.
Shadow-puppets wallow on the walls of ignorance
And the wise forget
That to be tainted
Is vice over gift.
The dawn crisps in.
We begin this waltz again;
“He who does not know Him
Is absent from Him who is
Everywhere present.”
The beautiful blonde with baby blue eyes
Cringes and cries,
And asks,
“Where are we?”
Forgive us, Father,
We’ve ceased to be.
Philosopher’s Guide: Gödel’s Incompleteness Theorem
April 7th, 2009
Introduction
There are few celebrity mathematicians. Naming one is quite difficult for those who are not students of mathematics. Newton and, Einstein often come to mind, but both are remembered in high school text books for their developments in physics, not mathematics. Kurt Gödel, on the other hand, was and remains a popular culture exile. Gödel’s two incompleteness theorems contributed more to the development of formal logic than any thinker since Aristotle, yet most undergraduates can name Aristotle.
Kurt Gödel was a complex figure in the history of mathematics for the mathematical and philosophical reach of his work. The two great theorems that redefined the scope of mathematical logic were certainly born of both great genius and great intrigue. Gödel was an enigma of a man whose reclusive tendencies garnered an air of wonder. He could often be seen walking to and from Princeton’s Institute for Advanced Study with the only man who could empirically be called his friend, Albert Einstein himself. To all but Einstein, Gödel remained a face of personal obscurity.
Gödel revolutionized mathematics similar to the way Einstein revolutionized physics. Gödel’s two incompleteness theorems can be stated like this, (i) “In any formal system adequate for number theory there exists an undecidable formula— that is, a formula that is not provable and whose negation is not provable.”
and (ii) “[T]he consistency of a formal system adequate for number theory cannot be proved within the system.”
These two brief statements disturbed the stagnating waters of the field of formal logic in the early 20th century. One reason, among others, that these brief statements are so remarkable is how far their implications reach. What Gödel accomplished in mathematical logic extends beyond mere mathematics to the grander philosophical questions that have been at the heart of Occidental intellectualism since the Greeks. The incompleteness theorems speak to more than simply what kind of well formed formulas (or wff’s) are decidable within a sufficiently complex system, but they also speak to the even deeper issues of what things can be known. In this post I hope to explain the magnitude of Gödel’s Incompleteness through examination of the preceding intellectual atmosphere, a brief explication of the proof itself, and an evaluation of the the interpretations.
The Logos and The Legos
One ingenious aspect of the theorem is its self-referentiality. By this is I simply mean the proof’s ability to discuss its own properties. It is intuitive to consider an evaluation of a system as being outside the system itself, but for Gödel’s Theorem this intuition is not the case. A rather pertinent example of logical self reference is the famous and most ancient Liar’s Paradox. An instantiation of the paradox looks like this, “This very sentence is false”. To make the point of the paradox more clear, one need only ask the question, “is the Liar’s Paradox true or false?” If it is the case that the sentence is true, then it can not be the case that it is true, because it claims to be false. Moreover, if it is the case that the sentence is false then it can not be the case that it is false because if it is false then it must be true! We are compelled by both our intuitions and the strict rules of logic to draw contradictory conclusions. But, consider the negation, “This very sentence is true.” Even this non-paradoxical phrase irks many thinkers who ponder it. Self referentiality is a curious and unique device.
In formal logic paradoxes can be the key to success or they can undo very rigorous work. The renowned British philosopher Bertrand Russell experienced critique formalizing mathematics into set theory (in his work with Alfred North Whitehead Principia Mathematica) after the popularization of what has been called “Russell’s Paradox”. Russell’s Paradox is similar to the Liar’s Paradox in that it is self referential. Sets are, simply put, are abstract containers. The things they contain are called members. The members of the set of all counting numbers are just the counting numbers, and there are no members in the empty set. Some sets are members of themselves and some are not e.g., the set of counting numbers is not itself a counting number and is therefore not a member of itself. However, the set of abstract objects is itself an abstract object and is therefore a member of itself. Russell’s Paradox refers to the set of all sets that are not members of themselves and asks the question is the set of all sets that are not members of themselves a member of itself. Princeton philosopher Rebecca Goldstein responds,
[Russell’s Paradox] either is or it isn’t [true], just as the problematic sentence of the liar’s paradox either is or isn’t true. But if the set of all sets that aren’t members of themselves is a member of itself, the it’s not a member of itself, since it contains only sets that aren’t members of themselves. And, if it’s not a member of itself, then it is a member of itself since it contains all the sets that aren’t members of themselves. So it’s a member of itself if and only if it’s not a member of itself. Not good.”
At first glance it may not appear that one hiccup in the system is reason enough to disregard it, but the existence of a paradox is far weightier than other seemingly simple problems. By asserting a contradiction in formal logic any proposition follows. If it is the case that a logical system can prove anything then it can functionally prove nothing.
Anything follows from a contradiction because of the logical inference called modus ponens. Modus ponens states that in a conditional statement of the form “if p then q” (where “p” and “q” stand for any two propositions and q can be derived from p) the truth value of the condition is only false in the case where p is true and q is false which implies that in any case where p is false (as in the case of a contradiction) q will always be true and hence, the truth value of the conditional will always be true. Contradictions, however, need not always result in such dismal logical scenarios
Contradictions can instead be used to prove theorems. In symbolic logic the logical inference called “negation elimination/introduction”
makes this clear. In its most basic form negation elimination states that if one assumes some proposition p and a contradiction results, what follows is the negation of p in the form “~p”. Gödel uses the very same rule to prove Incompleteness, and is in this way very simple. Gödel was able to assume a self referential statement, namely, “This very sentence is provable with in this system” (a statement very much akin to the Liar’s Paradox) and derive a contradiction thus demonstrating its negation, namely “This very sentence is not provable within this system.” These tools of logical inference were the building blocks of Gödel’s proof.
Platonism and Priorities
Gödel’s world-view should at least be mentioned for the purpose of evaluating his work later. I would in no way insinuate that the implications of Gödel’s work could at all be reduced to a biographical study of the man, but Gödel held his views with such conviction that it seems careless a exposition of his work to leave them unmentioned.
While at the University of Vienna, Gödel underwent shifts in his major studies and, concomitantly, his priorities. Gödel entered university intending to study physics, before switching to mathematics. When he first decided to study math, Gödel was particularly interested in number theory largely due to his Platonist convictions. He remembered the Viennese lecture on number theory fondly as being “most wonderful lectures he had ever heard”. The course offered in number theory was an exceedingly popular class attracting upwards of 400 students. And number theory would suffice for a time, but Goldstein has noted that in a letter responding to the question, “Are there any influences to which you attribute special significance in the development of your philosophy?”. “Heinrich Gomp. [erz] Prof[essor] of Ph[ilosophy] of Vienna.”was Gödel’s full response.
It is interesting to note that it was not the most excellent lectures he had ever heard that he cited as being influential to the development of his philosophy, rather it was a class called “Introduction to the History of Philosophy”. It was in this class that Gödel first encountered the great name of western philosophy, Plato.
It is no understatement to say that Gödel was enthralled with Plato’s philosophy and that it did not take long for the young mathematician to be convinced by Plato’s arguments. Goldstein agrees,
“I think it is fair to say, however, that like so many of us Gödel fell in love while an undergraduate. He underwent love’s ecstatic transfiguration, its radical reordering of priorities, giving life a new focus and meaning. One is not quite the same person as before. Kurt Gödel fell in love with Platonism, and he was not quite the same person as he was before”
It is not an uncommon occurrence to find a mathematician who holds to some form of Platonism. In fact, it seems that the field of mathematics is where the notions of Plato’s philosophy seem to have steadfastly endured. It is not a surprise. Plato was very mathematically inclined and desired his students to be as well. In Plato’s famed Academy, he had engraved above the entrance, “Let no man enter herein who has not first studied geometry.” Plato put a special emphasis on the study of mathematics for the comprehension of his paradigm.
When the term “platonism” is used in the context of mathematics it generally means a system of beliefs that presumes the properties of and derived from mathematics obtain an objective, eternal, non-physical or metaphysical reality equally real (if not more real) than the presently seen physical reality. A mathematician who is simultaneously a Platonist, then, generally believes that this thing called “mathematics” is akin to physics or any of the other natural sciences in that its students discover the properties which are independent of our mind and existence. Math, Plato believed, exists whether you believe it does or not.
Of course, Plato himself, did not stop at mathematics. Rather, Plato viewed the existence of mathematics as a kind of proof or evidence of his broader philosophy of abstraction. Plato called these abstractions “Forms”, the greatest of which is the form of The Good. Broadly speaking, The Good is that which all instances of goodness participate in. Similarly with The Beauty, there exists beautiful things, but only insofar as those things resemble true Beauty.
Kurt Gödel strongly believed that what he contributed to mathematics was not simply a construct language dependent on human understanding, rather it was a discovery in the same vein as the discovery of a planet in the solar system or the discovery of a new species of animal. Perhaps the analogy from astronomy is more fitting as the Forms exist in a kind of heaven enjoying pure and untainted being.
The Proof
A full explication of the details of Gödel’s proof is far beyond the scope of this post. The majority of Gödel’s paper (1931) was devoted to those details. I will however explain how the proof makes use of what is called Gödel numbering, the implications of Gödel numbering, and how the proof is able to make metamathematical statements.
Recall that the crux of the proof rests upon its ability to evaluate the statement, “This very sentence is not provable within this system”. The system is able to accomplish this rather stunning feat using a technique that has come to be known as Gödel numbering. Logical systems such as Gödel’s use a syntax of symbols governed by strict rules for how those symbols relate to one another in order to represent logical relationship. The symbols are strictly meaningless. The hope is that in using arbitrary symbols and rules for those symbols, one can create a system that makes no appeals to the ambiguity of intuition or self evidence. Attempting to explain every proposition in the vernacular is clumsy and may very well require an infinite list of lexical definitions to eliminate vagueness. However, it seems that no matter how precise the wording of a proposition may be in a natural language they can always be a semantic objection to terms, connotations, and other convolutions. Symbols however can not be argued with if they are arbitrary.
These symbols in some sense are used to represent logical relationships. Gödel numbering works to assign each symbol a number as shown:
The logical system that Gödel created with this numbering system is a formalized calculus called “PM”. It may seem contradictory for PM to use symbols that are meaningless in hopes to create meaning. Gödel scholars, Nagel and Newman anticipated this reaction in their explication of the proof and responded,
“[T]he reader might well wonder what earthly sense it makes to devote a column to the “meanings” of these supposedly meaningless symbols. Are we not speaking out of both sides of our mouths? The answer is that we are walking a subtle midway path between truly empty signs and truly meaningful ones . . .”
Nagel and Newman go on to explain that the right most column shown above presents the conventional meanings that are usually associated with their corresponding symbol, but the symbols themselves are, in fact, meaningless. As for deriving theorems from the symbols, one need only apply the rules properly and in this way the actual “meanings” of the symbols is arbitrary. What then is point is showing their usual meanings in the right most column? The usual meanings are given because formalization of calculi is meant to be as consistent with usual interpretations as possible so that a connection between the meaningful (interpretations) and the meaningless (symbols) can be established.
With Gödel numbering in place we will now be able to see in an elementary way how metamathematics can be derived. Consider the sentence,
(i)∃(x)(x=sy)
or
(ii)(0×ss0)=0
This sentence can be interpreted, “There exists some individual x in the universe of discourse such that, x is the immediate successor of y.” Next we examine the Gödel numbers as analogues of the sentence. For the above formula the Gödel number would be:
(i.a)48119811579
or
(ii.a)861377656
This is an example of how Gödel numbers can be assigned to wffs, series of propositions, and proofs (which are simply particular kinds of a series of propositions). The above numbering process is not identical to the numbering that Gödel himself used in 1931, but it suffices to demonstrate the idea behind numbering.
Examining how Gödel devised his system reveals the gravity of his proof. To simplify, every wff in the system bears a very particular arithmetical property. This implies that each wff has a certain relationship to all other wffs via the arithmetical properties. The relations between wffs however, is also expressible with in the system. This is what gives PM the ability to refer to itself. The system can express that some particular wff1 is factor of wff2 or that it is the nth successor of wff3, all of which can be expressed either in the language of the system or language of arithmetic. It follows that the meta-statements and the statements themselves “collapse into one another”.
Gödel devised PM in such a way that all proofs have a certain property. For our simplified example let us say that all of the proofs correlate to even numbers. This provides a relationship between numbers that allows us to show that all and only theorems have the property of being even. “You can see where we’re heading: toward arithmetical propositions expressible in the system that also speak to the issue of their own provability within the system.”
The final step of the proof is to create a proposition such that if the proposition is true then it is unprovable. This is achieved by creating a category within PM that designates only provable statements. Let “Pr” be the notation for provable statements. In order for there to exist such a category in PM, Pr must be a kind of arithmetical property that is the property of all and only provable propositions. In the syntax we might like to say that x is provable within the system (or stated otherwise, x has the property designated by Pr), we could simply state Pr(x). It follows that if some proposition p when expressed in Gödel numbering (let GN designate a proposition’s expression in Gödel numbering) such that GN(p), either will or will not be provable in the system. If it is, we can express that the statement is provable as Pr(GN(p)). If it is not we negate it as ~Pr(GN(p)).
Next we ought to take a moment to examine the statement Pr(GN(p)). Pr(GN(p)) (i.e,., proposition p as rendered in Gödel number is provable )is true if and only if p is provable. It is, in fact, a metamathematical statement. What Gödel accomplished with this expression was the ability to state within his formal system “This very sentence is provable within the system” From a version of this statement, Gödel (through rigorous logic which will not be reproduced here) demonstrated how to construct a statement that is true because it is unprovable. This was the first Incompleteness Theorem.
Give It To Me Straight, Doc! I Can Take It
The implications of only this first theorem are as diverse and unique as the mind that created it. The presupposition that Gödel, himself, had when he presented his proof in 1931 was Platonism. So it is fitting that the first question we ponder is whether or not we believe Gödel’s Incompleteness is as real (if not more real) than our every day experience. Given a kind of naïve realism about the external world what can we infer about not only Incompleteness, but also mathematics in general.
There seems to be a strong case to be made that mathematics will be real whether we are conscious of it or not. After all, Gödel has been dead for more than 30 years now, yet it seems like his proof is still true. Of course one could argue that it is in the minds of other mathematicians who have studied it since. Plato’s own arguments from recollection have never seemed to me to be sufficient. But the platonist has set for himself the difficult task of arguing for would happen in the instances no mind is capable of asking “What would happen?”.
Then there is the opposing view, that mathematics is simply a constructed language of the mind. It follows from this view that mathematics would simply not exist as we know it without the human mind. And by “exist as we know it” I am, of course, not simply referring to standard methods of notation as we have constructed it, but the “truths’ of mathematics as we generally accept them. The anti-platonist, if you will, is arguing for the same, strictly unverifiable, conjecture that assesses a possible world with which we have experience.
Both views seem to appeal to our minds as being sufficiently rational (and rationality having some necessary connection to reality) to be able to, at least, infer an answer. But the grounds for this presupposition seem dubious at best. We find ourselves posing the same sorts of questions as the skeptic, which is forbidden, or else we must resort to question begging in order to be able to hold either platonist or anti-platonist convictions. I remain personally agnostic on the issue.
Regardless of whether or not one takes the platonist position or not we are certainly left with a sense of urgency as the universe that includes Gödel’s Incompleteness is the universe in which we live. And, there is still the issue at hand, namely what does Incompleteness mean. The answer to that question may be as paradoxical we might have expected. There are those who would say that Gödel’s proof demonstrates that human beings will always be superior than machines. A machine, no matter how complex, only uses the rules the system that governs it. Incompleteness then will imply that there will exist truths that the machine can not prove and in this way the machine is insufficient. So perhaps it is the case that Humans will always know more than machines. But, as Goldstein points out,
“Of course there is no proof that we know all that we think we know, since all that we think we know can’t be formalized; that, after all, is incompleteness. This is why we can’t rigorously prove that we’re not machines. The incompleteness theorem, by showing the limits of formalization, both suggests that our minds transcend machines and makes it impossible to prove that our minds transcend machines.”
So we may not be machines, but we certainly can not prove that we are not. Does this leave the question strictly open? Does Gödel actually help us solve our epistemological problems or has he just given more weight to the skeptic to ground us even more firmly in the epistemic quagmire that we as thinking beings find ourselves in? Platonists and others have, perhaps ironically, given us unverifiable answers. To fully grasp the weight of the question is a feat in itself. Gödel’s proof is certainly the most elegant and most intricately crafted piece of intellectual history I have ever endeavored to study. Yet, there is something ominous about it, something that inclined Gödel himself to believe that there is more to be examined than merely the unknowable.
Philosopher’s Guide: Quantum Mechanics
April 7th, 2009
Experimenting with quantum materials has led to a number of unusual, unpredictable, and often “weird” results coupled with even weirder interpretations. However, the quantum facts are undisputed. The best illustration given by Dewitt is the classic “two-slit” experiment.
This experiment was designed to attempt to settle the question of whether quantum materials are waves or whether they are particles.
Firstly, Dewitt offers a brief and clarifying explanation of the difference between particles and waves. Particles are discrete objects, with reasonably well-defined locations in space and time”
A baseball is Dewitt’s example. A book, a frisbee, and backpack are also large size examples of how we might think of particles. Particles have particular properties as well. When particles interact with one another they might bounce off of each other or break into smaller particles.
Waves on the other hand do not have well defined boundaries in space and time. Consider a ripple in a pond or an ocean wave approaching the shore. The ripple or the wave is not located in one particular location in space. Rather, their exact location is spread out over a relatively large distance. Waves also do no interact in the same way as particles. They do not bounce off of one another or break up into smaller waves. Sometimes two or more waves will “cross paths”, so to speak, and cancel each other out. It may be the case that the two waves combine to become an even larger wave, or two waves may cross and be left completely unaffected by the interaction.
Particles and waves also have “very different experimental effects”.
Suppose that we “shoot” a steady stream of particles, in this case pellets or paintballs, towards two open windows and ask, “what kind of pattern will emerge on the other side of the windows?”. Intuitively we might be inclined to say that the paintballs that pass through the windows will accumulate in two areas beyond the window. This is the foundation for the two-slit experiment.
In the two-slit experiment an electron gun shoots a stream of electrons at a barrier that has two slits (as in the figure to the right). Beyond the two slits is a photographic paper that will detect the electrons.
If electrons are particles then we would expect to see two “piles” of particles detected on the paper. The piling up of electrons is called the “particle effect”.
Contrastingly, if electrons are waves, and we repeat the two-slit experiment, we would expect the slits to split the wave into two just beyond the barrier. Those two waves would interact with each other as they hit the photographic paper and would create an interference pattern of alternating dark and light bands. The interference pattern that would result on the photographic paper is called the “wave effect” (as seen in the figure below).
When the two slit experiment was conducted, it yielded a clear wave effect, i.e. the photographic paper had a distinct interference pattern on it. The same experiment was conducted again with only one slit open at a time, alternating open and close, but both slits were never open at the same time. When this second experiment was conducted the result was a clear particle effect. Here is the first indication of “weirdness”. The first experiment would indicate that electrons are waves whereas the second experiment indicates that electrons are particles.
In an attempt to better understand this perplexing phenomenon, a third experiment was conducted. This third experiment is identical to the first experiment with this addition, a passive particle detector is placed at each slit. Also, for this experiment, the rate at which the electron gun fires is decreased significantly. If electrons are waves, then both detectors should register a particle at the same time. If electrons are particles, then the detectors should only register one at a time. Recall that the first experiment yielded a clear wave effect. However, this third experiment yielded a clear particle effect. Also, note that the particle detectors are passive, i.e. we do not expect them to interfere with the particles at all. Rather, we expect them to simply note whether or not a particle is present. It is intriguing and confounding how, if at all, these detectors altered the electron.
The fourth experiment is identical to the first except in this instance drastically slow down the rate at which the electron gun fires. The electron gun will fire as to only emit one electron at a time. As this experiment was conducted only one electron appeared on the photographic paper at a time. But the electrons did not correspond to the two slits. Rather, over time, the electrons made a distinct wave pattern (as in the figure seen below).
The final two experiments use a photon gun instead of an electron gun. A photon gun emits photons or quantum units of light. These experiments will also use a beam splitter, a beam re-combiner, photon detectors, two mirrors and photographic paper to record the results (as seen in the figure below). For this experiment the photon detectors are turned off.
If the photons are waves then then the wave will be emitted toward the beam splitter, splitting the wave into two. One wave will continue straight toward the mirror while the other will be reflected down toward the beam re-combiner. When the first wave hits the mirror it will be reflected down towards the re-combiner. The two waves will then be reflected towards the photographic paper. Because there are two waves, the effect produced should be a wave effect. When this experiment was performed the result produced the wave effect.
The sixth and final experiment is identical to the fifth with one alteration, the photon detectors are turned on. With experiment 5 indicating that photons are waves we would expect that the photon detectors would register photons simultaneously, but, counterintuitively, the photon detectors register photons one at a time. Furthermore, when the detectors are turned on, the result is a clear particle effect. The experiments and their results have been summarized in the table below.
The results are troubling for a host of reasons. Perhaps the most intuitive of which is the notion that quantum materials seem to act both like waves and particles. Complementarity, typically associated with Niels Bohr, refers to effects such as the wave-particleduality (as seen in the figure shown below),
exactly like the kinds mentioned, in which different measurements made on a system reveal that system to have either particle-like or wave-like properties. As was previously discussed, waves and particles are very different from one another and it is difficult to understand what kind of world that would produce this kind of data.
These quantum facts have given rise to various interpretations. Perhaps the most influential of which is called the Standard or Copenhagen Interpretation (CI). The first tenant of CI is that there are no hidden variables, i.e. that quantum theory is a complete theory and does not need so-called “hidden variables” to make it more coherent with our current intuitive understanding of the world
. With the assumption of no hidden variables, CI attempts to explain unusual quantum data through with addition to mathematics called the Projection Postulate (PP). PP refers to the collapsing or reducing of a wave function.
CI suggests that “before a measurement, a quantum entity is represented as existing in a superposition of states” and is “ . . . represented by a wave function”
This means that the quantum entity is represented by a wave of probability that governs the multiple places in space that the entity can be measured. However, when a measurement is taken, as in the case of the experiments above, the superposition collapses, creating a new wave function. The collapse is governed by the mathematics of PP. The new wave created represents quantum entities a either detectors A or B.
Wave mathematics in addition to PP have been marvelously successful in predicting and retrodicting the quantum facts. In fact, CI can accurately describe the data observed over the past 70 years of quantum experiments.
There is a problem however when using CI to try and describe reality. Questioning where the electron is actually located the instant before it is measured is problematic. Under CI, one might have to assume that the electron is at both detectors A and B at the same time and then mysteriously disappears the instant before it is measured. “Advocates of [CI] generally take the view that there are no answers to the questions such as these. For example, we simply cannot say where the electron is really located before measurement.”
Similarly with an electron’s attributes e.g. it’s velocity, momentum, and spin, all remain unknown before measurement. This is Heisenberg’s Uncertainty (or Indeterminacy) Principle. Formally stated, locating a particle in a small region of space makes the momentum of the particle uncertain; and conversely, that measuring the momentum of a particle precisely makes the position uncertain.
This is not a typical kind of uncertainty, however. It is not the case that the particles in question have attributes but we are simply unaware of them.
According to [CI], the reason we cannot say what attributes a quantum entity has before measurement is not simply because we do not know what those attributes are. Rather, we cannot say what those attributes are because those attributes do not exist prior to measurement. There is no deep, independent reality consisting of objects with definite attributes existing prior to measurements of those attributes.
This is a far cry from the absolute-space and absolute-time intuitions of the Newtonian world-view in which matter is objective and independent of the human mind, where time passes in the same way everywhere. The conundrums of quantum theory leave much to be desired in the way of intuition. Nevertheless, the undisputed quantum facts require an interpretation insofar as we desire empirical science to describe the external world. CI casts light on many of these ambiguities and has revolutionized the way science views our conscious interaction relative to physical matter.
Philosopher’s Guide: Non-Euclidean Geometry
April 7th, 2009
This is the first in a series I’ve called the “Philosopher’s Guide”. Entries in this series will contain brief summaries of advanced topics in what I hope will be understandable and plain language. As a philosopher I hope this series will provide a foundation and a language for understanding difficult topics in academia.
Introduction
The affects of the development of non-Euclidean geometry are far reaching to say the least. Firstly, to successfully bring doubt to the propositions of Euclid in the 19th century was a feat in itself. It was formerly the case that questioning any of the postulates of Euclid was to question certain and indubitable truths. The thought that highly intelligent and intellectual people should somehow show that Euclid had a flaw of any kind was nigh unthinkable. As John Stillwell put it in his book, Mathematics and its History
Euclid’s Elements wielded “absolute authority, both as an axiomatic system and as a description of physical space.”
The two notions, (i) that Euclid’s proofs were logically certain, and (ii) that Euclidean geometry was a real mathematics of physical space, were common assumptions that underwent a vast revision with the rise of non-Euclidean geometry. If it could be shown that Euclid’s geometry had dubious foundations or that one or more of his axioms could be denied, and still yield a consistent system, then the basic assumptions about physical space would be in need of reform.
Origins
Three names commonly associated with the development of non-Euclidean geometry are Gauss, Bolyai, and Lobatchevsky. These men rigorously and independently from one another completed nearly identical work forming the the first theorems of non-Euclidean geometry. Morris Kline, in his work, Mathematical Thought From Ancient to Modern Times
, warns the reader that it was not solely the work of the these three men that birthed this new geometry, though they wrote the first substantial texts on the topic. Kline clarifies, “There is a common belief that Gauss, Bolyai, and Lobatchevsky went off into a corner, played with changing the axioms of Euclidean geometry just to satisfy their intellectual curiosity and so created the new geometry”
It is more realistically the case that the three intellectual giants had a long history of men preparing the way for them as they laboriously tinkered with the implications of denying (specifically) the parallel postulate.
The parallel postulate
states that if when one straight line L0 crosses two straight lines L1 and L2 to create interior angles α and β whose sum is less than π
then L1 and L2 meet on the side of L0 that contains α and β as seen in the figure.
The first implication is that if the sum of the angles α and β is greater than π, then the two lines will meet on the opposite side. The second implication is that if the sum of α and β is equal to π then lines L1 and L2 do not meet.
The major question for those probing Euclid for flaws was, where does this postulate come from. Is it derivable from the other nine axioms? Many great thinkers and mathematicians e.g. Lambert, Schwikart, Taurinus, et. al. worked on the above question attempting to prove its validity or derive a contradiction from its negation
. It seems that none of these great thinkers were able to derive a definitive answer to the satisfaction of the mathematical community or to themselves
. What came out of their inquiries however, laid the foundation for Gauss, Bolyai, and Lobatichevsky’s work.
Certainty and Realism
What is a straight line? Is a curve a straight line? What follows if the the parallel postulate is denied? These questions and more were the starting places for the development of non-Euclidean geometry. The figure below depicts different ways that the parallel postulate can be denied.
These models demonstrate how powerfully a simple conceptual shift can alter the entire frame of reference for mathematical systems. But, as Kline was careful to note, it was not for the mere satisfaction of their curiosity that great thinkers have altered the assumptions of a system. Rather, it was the ambiguous and undefined nature of the parallel postulate that warranted its denial.
Gauss was the first, according to Kline, to achieve the first major conceptual step towards non-Euclidean geometry. The parallel postulate enjoyed millennia of ataraxic dominance. Those who realized that it was not to be derived from Euclid’s other nine axioms were thus free to deny it, creating revolutionary new geometrical systems. And these men fell in line with many similarly great thinkers. Gauss stands alone however, as he was the first to abandon the interpretation of Euclid’s geometry that necessitated its relationship to physical space
. Denying the parallel postulate opens up entirely new possibilities for consistent geometrical systems. Denying that Euclid’s geometry is ‘real’ gives way to entirely new possibilities concerning the philosophical foundations of geometry itself.
Gauss classified the so-called truth of geometry with studies such as mechanics
, i.e., truth as verified by experience (as opposed to its former lofty and prestigious verification by a priori Reason). This paradigm shift began to dismantle the logical certainty of Mathematics. Until this point in history to say that something was as sure as Euclid’s geometry was to say that it was indubitable and held the same class of certainty as the cogito. Philosophers and Mathematicians alike were now forced to consider geometry as one strain of possible truths. Perhaps there exists a possible world in which the interior angles of a triangle do not sum to π. Perhaps that possible world is our world. Imagine that it becomes the case that a gifted mathematician could demonstrate that their exists some rational number that is precisely equal to the square root of two. “That can not be.” the skeptic might reply. “We have sophisticated and elegant proofs that deny that such a rational number exists.”
Where then, can one draw the line between what is simply reasonable and what is necessarily true? Non-Euclidean geometry left the mathematical community bereft of its certainty. The ‘real’ was now becoming ambiguous. Mathematical thinkers were not lost forever. Contemporary thinkers have created models of physical space that utilize non-Euclidean geometry elegantly. What the development of non-Euclidean geometry shows is not that our epistemological foundations are dubious, rather, that our assumptions are neither more or less than splendid dogmas subject to scrutiny.