November 27, 2008
Materialism and the Concept of Humanity
It’s been far too long since I’ve posted something here- the useful must come before the useless.
My thoughts are the result of listening to a recent YPU debate about the danger of biotechnology to some notion of humanity, and I was struck by the seemingly universal assumption that “humanity” is something good, great, wonderful, true, and whatever other adjective you’d like to throw at it. There was precisely one good speech all evening. People are downright too into being human to be critical about any notion of humanity.
Humanity as anything more than a species distinction is a lie. At the heart of “eliminative materialist” philosophy is the idea that our intuitive notions of our own psychology must not be taken at face value. Through the process of evolution, we have been designed to view ourselves in a way that maximizes our probability of reproduction and survival, and there is no a priori good reason why true self-knowledge is an evolutionarily useful trait. What we believe makes us happy may not make us happy. Furthermore, our conception of our personal identity need not be connected to reality.
Eliminative materialism can be viewed as an update of the empiricism sometimes associated with Hume. The most susinct statement of this older philosophy can be seen in the statement: ‘Nothing is in the intellect which was not previously in the senses.’ Eliminative materialism takes this already austere conception and brings it one step further- not only is sensory experience the sole charecterization of the mind and even conciousness, but sensory experience is in some sense isomorphic to the chemical structure of the brain. On a fundamental level, all of our most powerful feelings are in a reductionist persective “only” the dynamics of brain processes. This is not to devalue the profundity of conciousness- that would be analogous to calling a house merely a pile of bricks or a symphony merely a series of changes in air pressure. However, this reductionistic perspective can lead us to some counterintuitive insights.
Nowhere in the dynamics of a physical system is there space for any immaterial essence that defines humanity. The fact that we are ourselves humans does not change the fact that we must be highly skeptical when studying humanity- subjective experience is one of several sources of information about human beings. We must proceed from the perspective of what Daniel Dennett has labeled “heterophenomenology”- the phenomenology of another, not oneself. To understand ourselves, we must move outside ourselves and apply the same sort of methodology we would use to understand any other system. The feeling of something profound which “makes us human” must be viewed skeptically, and we must attempt to connect any such conceptions to the cold, hard facts of sensory perception.
Dennett’s article “Quining Qualia” provides an series of what he calls intuition pumps that question our idea of any kind of subjective experience beyond the sensory. Can we grow to enjoy the taste of asparagus without changing the taste? Is phenolthiourea (which tastes disgustingly bitter to about 3/4 of people but is tasteless to others) bitter or tasteless, and is this a propertty of phenolthiourea or a property of our senses? To push his examples towards the foundations of our concepts of humanity, can we feel the the subjective experiece of joy while actually not being joyful? If we then extend this reductionistically (and thereby enter the realm of the counterintuitive), can we feel happiness without our brains being filled with dopamine, oxytocin, and/or serotonin? All of the seemingly natural concepts and distinctions we make when pondering our own conciousness naievely are like quicksand when these sorts of questions are properly asked.
Thus, to return to the initial inspiration for this post, our ability to shape our conciousness through biotechnology will clearly be an assault on the concept of humanity. If we change our subjective sensory experience, the subjective notion of being human will be on the list of traits which might be modified. This freedom in and of it self if universally comprehended easily will falsify any folksy or humanistic theory of conciousness. Indeed, in certain quarters, it already has been. A conversation with someone who has experimented seriously with hallucinogens is all the evidence you need that serious modifications of our sensory experience can profoundly change our sense of humanity. It is misguided and arrogant to say that the sensory reality we live in is the “true” one. If we are to approach this from an empirical/materialist perspective, sensory experience IS reality. There is no room for anything more.
July 8, 2008
The Principle of Anodine Choices
I recently read Alexander Grothendieck’s “Equisse d’un Programme”- an extremely insightful summary of one of the most interesting mathematical research programs of the century. On page 29, he presents an extremely interesting general principle of mathematical research, which he calls the “Principle of Anodine Choices.” In his words, “When for the needs of some construction of a geometric object in terms of others, if we are led to make a certain number of arbitrary choices along the way, so that the final object appears to depend upon these choices, and is thus stained with a defect of canonicity, then this defect is indeed serious (and to be removed requires a more careful analysis of the situation, the notions used, and the data introduced). Whenever at least one of these choices is made in a space which is not “contractable” i.e. a space whose invariants are non-trivial, and that this defect is on the contrary merely apparent, and the construction itself is “essentially” canonical and will not bring along any troubles, whenever the choices made are all “anodine” i.e. made in contactable spaces.” (translated from french and hence a bit unnatural sounding)
To me, this gets at a very fundamental aesthetic point in mathematics- that the objects we construct should embody the structure of the objects we use to construct them. If we must introduce arbitrary extra data into our constructions, then these constructions are mathematically unnatural, and we must avoid them however we can. From an aesthetic point of view, we must express our concepts in a language which does not constrain them with more structure than is necessary to define whatever concepts we wish to examine.
I would like to consider an example-
Consider the notion of continuity of functions from R to R. A function f: R –> R is continuous if for every real number x, the limit as y approaches x of f(y) equals f(x). When we write out the definition of limit, we get a mess of deltas and epsilons telling us that for every interval of length epsilon around f(x), there is some interval of length delta around x that maps inside of it. Thus, our definition of continuity refers to a series of quantitative bounds on distance in the euclidean metric.
Despite this, the definition of continuity is really not fundamentally related to any quantitative notion of distance on the real numbers. We prove the following:
THEOREM: A function f: R —>R is continuous if and only if the preimage under f of every open set is an open set.
PROOF: Suppose f is continuous. Let S be an arbitrary open set in R. To show that the preimage of S is open, we must show that every point in the preimage of S is an interior point in the preimage of S. If the preimage of S has no points, it is the empty set and is thus defined to be open, so we can assume the preimage of S contains at least one point. Fix a point p in the preimage of S. Because f(p) is in S, and S is open, an interval I of some length epsilon about f(p) is contained in S. Because f is continuous, some interval of length delta about p is mapped into I and thus has its image contained in S, so p is an interior point. Thus, since S is arbitrary, we have shown that the preimage under f of every open set is open.
Conversely, suppose the preimage under f of every open set is open. Fix a point P in the image of f and consider an interval K of length epsilon around P. Since intervals are open sets, we know that the preimage of K is open. Since P is in the image of f, some point z in R must map under f to P. Since P is in K, z must be in the preimage of K. Since the preimage of K is open, some interval of length delta about z must be contained in the preimage of K, so we have an interval about z of length delta that maps into an interval of length epsilon about P. Thus, f is continuous at z. Since every point in the domain of x maps to some point in the image of f, and P is arbitrary, this shows that f is continuous. QED
This proof shows that the quantitative bounds we find in terms of the euclidean metric to prove that a function is continuous are really not necessary. Any other metric which induces the same open sets as the euclidean metric will also define exactly the same functions to be continuous. Any map which puts the open sets of R in bijection with the open sets of some other space (called a homeomorphism) will form a natural identification between the continuous functions in R and the continuous functions in the other space. It is only the structure of open sets (called topological structure) that matters in any notion of continuity. Thus, it is much more natural to define a function f to be continuous if the preimage under f of every open set is open, and throw away our deltas and epsilons. Their mentioning brings in a concept which fundamentally has nothing to do with the concept of continuity. The choice of a specific metric is an “anodine” choice- its relation with continuity is an illusion which a better definition of continuity will destroy.
May 17, 2008
Negative Income Tax vs. Welfare
The argument that we cannot improve upon welfare has come up enough times that I feel like I need to argue for the natural alternative: the negative income tax (NIT).
There have been a number of proposals that I would consider to fit under this heading, most famously Milton Friedman’s proposal in “Capitalism and Freedom.” For the purposes of clarity, I define a NIT proposal to satisfy:
1. People who have no income receive something analogous to welfare, but smaller.
2. The amount of money that a poor person receives varies continuously with their income. That is, they gradually lose their welfare check as they make more and more money. Beyond a certain point, this decreasing payment becomes zero and then becomes the income tax.
If economic theory is remotely a good approximation of human behavior, it seems to be a no-brainer to me that this will lead to higher income among poor people than standard welfare. In the current system, welfare recipients have a disincentive to work- it is quite possible that with an especially shitty job, they will make less money in the workforce than they do on welfare. Thus, some people who would work if welfare did not exist will now not be working. By making payments gradually reduce as income grows, this disincentive becomes many times weaker. No matter what, having a job will increase your income. Simple, no?
Given that pure reasoning in an economic framework is not 100% reliable and might sometimes lead to thinking tinged with the libertarian bias inherent in economic theory (as much as I label myself as a libertarian, I feel that it is important to recognize that the intuition behind even very scientific models might have an ideological bias), I think it is good to see how this hits the data in the real world. The Earned Income Tax Credit functions in a manner similar to how a negative income tax would work. Here is an empirical paper on its effects http://harrisschool.uchicago.edu/About/publications/working-papers/pdf/wp_07_20.pdf .
Given how both theory and data seem to make the NIT seem like a better thought out choice than welfare, why is this not a political issue? How can people spend so much effort arguing back and forth about political ideologies when more good could come from simply improving the policies we have, regardless of the ideologies that brought them there in the first place?
May 10, 2008
Against Logical Atomism, part 2
1.2 A statement made in language does not directly express a truth about the world.
1.2.1 Since any formal definition of a word must use only other words, the process cannot “bottom out.” One must either accept some undefined terms or face infinite regress.
1.2.1.1 A formalized language, in which every term is either undefined or defined precisely through undefined terms, can thus make statements only about its undefined terms. A statement about a defined term reduces to a statement about undefined terms.
1.2.1.2 Undefined terms are in and of themselves meaningless. They are placeholders.
1.2.2 The assignment of truth values to statements in a formal language makes a claim as to how our undefined terms behave structurally.
1.2.2.1 By assigning truth values in a manner consistent with our some notion of deduction, we create a language which embodies a form of logic.
1.2.2.2 A language which embodies a form of logic which has not been contradicted by scientific investigation is quite useful. Such a language has the possibility of describing the world.
1.2.2.3 Any abstract structure is best described in a language which has a similar notion of deduction. Newtonian mechanics can be expressed in the language of calculus. Quantum mechanics can be expressed in the language of functional analysis. Quantum mechanics does not make Newtonian mechanics useless. Newtonian mechanics still makes coherent assertions within its language. The question of finding the natural language in which to speak of a certain topic is essential to any precise communication.
1.2.3 The properties a language guarantees about its undefined terms allows for meaning to be ascribed.
1.2.3.1 If one observes a system exhibiting properties that are mirrored by the undefined terms in a language, one can in some sense “perform the isomorphism” between the two and use the language to speak about the objects.
1.2.3.2 The meaning of a language is therefore not inherent to the language. It comes from our ability to use the language to mirror something else. The language does not mirror something else in and of itself.
1.2.3.3 Thus, we find a flaw with the picture theory of logic. One could say that logic is a picture, but it is wrong to say that it is a picture of the world. A picture is an object itself. By saying that one object or structure represents another, we must be saying that one can find an isomorphism between the two. It is only through perceiving the relations of a picture to the world that we can speak of the picture being any sort of representation. It is not inherent.
Against logical atomism.
1.A fact in and of itself is not an atomic fact, just as a vector is not in and of itself a basis vector. To say that a proposition is a truth function of elementary propositions is thus a mistake. One must say instead that any collection of facts which provide axioms from which the world’s truths can be derived guarantees the truth values of a collection of statements, which we can call propositions.
1.1 The study of the possible mappings between different representations of the same logical space can thus be considered the fundamental research program of mathematical logic.
1.1.1 A deduction is a mapping with the property that if every statement in the mapping’s domain is true, then every statement in the mapping’s range must be true.
1.1.2 We should not view a logical space as merely a collection of statements. To define a logical space, we must also state exactly what qualifies as a deduction. Like Klein did with geometry, this approach naturally allows for the problem of classifying all possible logics.
1.1.2.1 In some sense, this is precisely Klein’s erlangen program. A geometric property, like a logical property, is defined merely by its invariance under a transformation in this approach. The words “geometric” and “logical” are undefined terms.
1.1.2.2 Frege’s program of deriving mathematics from logic, even if sucessful, would not have shown logic to be any more fundamental than any other part of mathematics. We could very well be able to derive logic from algebra as well as algebra from logic. One example of this in the real world is the application of Topos theory to studying logic.
1.1.3 A basis of atomic facts for a logical space combined with rules of deduction determines exactly what statements are in a logical space. Likewise, a basis of atomic facts combined with the entirety of the statements in a logical space gives us restrictions which our notion of deduction must satisfy.
1.1.4 Our experiences give us evidence that a collection of statements about the world is true. The image of these statements under all deductions constitutes precisely what we can know about the world.
1.1.4.1 Different definitions of deduction can lead to the extrapolation of different logical spaces, different worlds, from the same experiences. The determination of which definitions of deduction will lead to the extrapolation of worlds that are not contradicted by our future experiences is one of the tasks of scientific theory.
April 18, 2008
Some Pseudoscience
I happen to be currently in a wondrously boring economics class in the dullest sense of the word. We have our nice little model where consumers maximize utility and firms maximize profit, trade is perfect and instantaneous, and information is perfect known by all parties. As much as I’m a fan of economics as a useful tool for understanding the world, I would like to call bullshit.
From a historical perspective, the entire theory of general equilibrium on which neoclassical microeconomics is founded seems, well, lifted from more respectable branches of science. Two of the big foundational books in general equilibrium, Samuelson’s Foundations of Economic Analysis, and Debreu’s Theory of Value, are developed more by analogy than observation. Samuelson’s model was the physics/chemistry of Josiah Willard Gibbs. Debreu’s was the mathematics of the mathematicians working under the pseudonym Nicolas Bourbaki. Thus, we see Samuelson applying Le Chatelier’s Principle about chemical equilibrium to studying economic equilibrium, and Debreu arguing that the fundamental theorems of welfare economics can really be viewed as results about topological vector spaces. Much of the field of “blackboard economics” is the direct result of Debreu’s work. Needless to say, I much prefer the legacy of Samuelson over that of Debreu, especially given his work on theories such as revealed preference which attempted to make economics more falsifiable.
While my criticism does not extend to economics as a whole, I am simply stunned by how pseudoscientific it seems in my economics class. We have never looked at a single bit of data the entire time. We have not discussed alternatives to the general equilibrium model, and all of the calculations we make on problem sets are for specifically those utility functions which make the results look pretty. One of the greatest bullshit detectors of all time, Richard Feynman, says in “Cargo Cult Science”
Details that could throw doubt on your interpretation must be given, if you know them. You must do the best you can — if you know anything at all wrong, or possibly wrong — to explain it. If you make a theory, for example, and advertise it, or put it out, then you must also put down all the facts that disagree with it, as well as those that agree with it. There is also a more subtle problem. When you have put a lot of ideas together to make an elaborate theory, you want to make sure, when explaining what it fits, that those things it fits are not just the things that gave you the idea for the theory; but that the finished theory makes something else come out right, in addition.
Theoretical economists could do well to start following this advice.
March 22, 2008
Undefined Terms: A Defense
The Reactionary Epicurean, as far as I can tell the only other mathematically inclined Yalie on the blogosphere, has posted a very nice rebuttal; of my first post, “Undefined Terms.” (While you’re on his blog, check out the video about Mobius Transformations for a nice bit of beautiful math.) While I intended the post primarily as a semi-ironic statement criticizing the possibility of my blog being as logical as I would want it to be, I still would like to counter a few of his claims.
The Reactionary Epicurean makes the assertion: “I’d submit the possibility that the form of communication has an effect upon the act of perception itself, leading us to the conclusion that Nietzsche, Hofstadter, and Marshall McLuhan were right all along: “The medium is the message.”" This is one of the primary reasons I distrust poetry, for example. Indeed, our perception is altered by their utterly non-deductive nature. As a form of art unto itself, I might enjoy poetry, but I consider it dangerous to look for any “truth” therein. In a world where people were cognizant of language’s power to manipulate and consciously avoided using it, I posit that we would be rid of many stupidly destructive ideas. (Religion is the best example, in my opinion.)
I also would like to dispute his simplistic gathering of Nietzsche and Douglas Hofstadter in the same camp. In the introduction to Godel, Escher, Bach, Hofstadter gives a very good one sentence summary of his views on language- “Meaningless symbols take on meaning in spite of themselves.” I certainly agree that, as Hofstadter also says, “In form, there is content.” However, I do not think this gives imprecise, flowery language a green light. To me at least, it provides evidence of the terrifying powers of a society which communicates illogically.
March 18, 2008
Answering Meaningless Questions
I used to think that a question that was empirically meaningless was simply not worth asking. One cannot articulate a well-defined answer without descending into a slew of vague buzzwords, so I did not think there was anything to this. Such questions include the “meaning of life” “nature of love” “what it is to be a good person” and essentially any question which we respond to primarily in an emotional way. However, I think there is a place for such questions even in a logically precise philosophy. The catch is that one does not answer these questions with a statement or proposition- one must answer with an action, habit, or way of life.
At the risk of appearing obsessed, I’d like to return to Wittgenstein’s “Tractatus Logico-Philosophicus.” One essential point in this book is that what language communicates cannot be referred to with language. A series of propositional statements each have their own logical sense, but they communicate in another way as well. The propositions also illustrate something, and this something cannot be referred to directly in language. I think this distinction is at the heart of the merit of an illogical, unanswerable question. The questions do not entail a precisely formulated statement which refers to something, but they can have the ability to illustrate something indirectly. Asking a “philosophical question” is a suggestive tool to make us reflective about the world, and I think we must respond to them accordingly- we must respond through the world.
Thus, if someone asks me “what is the meaning of life,” or something else equally cheesy, I think the best way to respond is to simply live. Given that “meaning” is itself a notoriously ill-defined term (again, that which language illustrates it cannot describe), I think answering such a question directly reduces to a useless play with words. That which we consider “meaning” is not something which we can logically deduce, it is in some sense an emergent property of living well.
From this point of view, I think the philosophy of existentialism is rather clearly misguided. We cannot sit around and consciously choose what we consider meaningful. To do so reduces to little more than trying to supply a definition to the word meaning, and such introspective philosophical techniques often reduce to little more than playing with language. That which we can speak of we must speak of precisely, and that which we cannot must be approached by other methods. To live in a fulfilling way, we must simply live in a fulfilling way, there is nothing more to it. Our conscious mind does not have complete power to decide what our subjective, unconscious mind values (I’ve tried), even though we may have some ability to consciously manipulate how we think. To imagine that our perception of meaning is totally our choice is to mistakenly overestimate the power of our ego. We are a product of our unspeakably ill defined thoughts just as much as our unspeakably ill defined thoughts are a product of us.
March 16, 2008
Formal and Informal Model Building (part two)
At the end of my last post, I pointed out that model building is inherently not metaphysical. Because the only strict criteria for the success of a model is that it fits empirical data, it says nothing about how the world “got that way.” From the stark, positivistic definition of the word “meaning,” I would like to argue that anything more is meaningless.
For the rest of this post, I will define a statement to be meaningful if it asserts something empirical about the world. The statement “The earth is round.” is meaningful because one can test this assertion by viewing the earth from space (assuming your definition of earth and round are the conventional ones). The statement “the earth is beautiful” is not directly meaningful, since there is no obvious definition of the word “beautiful” which allows one to conduct an experiment to test this assertion. Any type of question with the word “why” in it cannot have a complete, meaningful answer. For example, conservation of energy is “why” a pendulum never swings higher than the point from which it was dropped, but one is then left with why energy is conserved. Empirical tests can only see if something does or does not happen- data does not provide a narrative. To answer a why question scientifically, one eventually must state the fundamental, unproven assumptions of a model.
It is precisely for this reason that I think we can do nothing better than build a model of reality. Any meaningful theory will tell us how to interpret data, so we cannot even in principle be certain that we are not making simplifying assumptions in how we measure something. Even outside the realm of science, we rely on informal models. Marxism provides a (falsified to the extent it is falsifiable?) model of how history develops, as do many other political ideologies. None of the narrative portions of such political ideologies are meaningful in my strict definition of the word. Even in every day life, we informally model the behavior of others and act according to our predictions. Such models clearly have unfalsifiable portions (what precisely does it mean to have a “bad day” for example), which must on some level be meaningless. Thus, in all endeavors, I believe that model building is the best we can do. We can use our models as a way of thinking about the world, but it is meaningless to say that our models have anything to with “how things really work.”
March 11, 2008
Formal and Informal Model Building (part one)
Given that I am strongly considering a career in academic economics, I am very much a believer in mathematical modeling. Economic interactions are really complicated, and I personally don’t have the intellectual power to even attempt to understand how the economy “really is.” Thus, if I want to develop an understanding of something, I must make simplifying assumptions. Simplifying to the point of triviality (kind of like building a model :-0 ), this is my outline of how we build a model of something in the world.
1. We assume there is a “real world out there,” and proceed to make observations.
2. We perhaps notice some regularities in how stuff happens.
3. We choose some fundamental assumptions (useful lies, one could say) which allow us to build a theory.
4. We derive the consequences of our assumptions to make predictions about how the world works.
5. We do, or tell some empiricist to do because we are lazy theorists, an experiment or look at a data set and see if our ideas cut the mustard.
From this process, we get a mathematical model (yay). A falsifiable model will spit some numbers out which are supossed to be measurable in the real world, which we can then test. The model will then enter one of two categories- possibly right (since it has not yet been falsified), or wrong. As time goes on, this process of falsification allows us to keep the level of bullshit in our models pretty low. A falsifiable model has to be right in some very general sense of the word in order not to be thrown out. However, this does not mean that a not yet falsified model describes the world as it “actually is.” For example, no sane economist would ever actually tell you that there is some time independent thing called a utility function or objective measure of happiness which your behavior necessarily maximizes. One could only say that assuming that people maximize utility allows one to make useful inferences about the world.
To me, this is the largest philosophical stumbling block in the process of mathematical modeling. I could very well model, say, continental drift by the actions of the tooth fairy and get a falsifiable model which makes good predictions. From the austere perspective of mathematical modeling, such a model is exactly as good as any other which makes the same predictions. From a positive, empirical point of view, a model reduces to simply making assumptions to make predictions. Any assumption we make must in some sense be false, as any simplification of reality is, but this does not matter. There is no metaphysics in mathematical model building! In part two, I will hope to extend this claim and thereby touch on some fundamental issues in my logical positivism.
