INTERVIEWER: I've been thinking about your theory of knowledge since our last meeting, and I have some questions for you.
LOGICAL POSITIVIST: Certainly.
I: I don't expect a complete and comprehensive analysis of Logical Positivism -- I don't even want to parse out the "official philosophy" and the non-official offshoots from the broader influences and legacies --
LP: "Emanations radiating from the penumbras" and all that -- I understand. You want the -- "core values", so to speak, not a list of qualifying members.
I: So to speak. You state that the basic structure of Logical Positivist knowledge consists of premises -- verified by observation -- from which logical inferences are made, generalizations formed, and conclusions drawn. "Sound" arguments consists solely of true premises and valid inferences -- inferences which preserve truth value -- is that correct?
LP: It is. This is the highly efficient and fruitful culmination of the tradition of "getting the facts" and forming universal laws about the nature of reality. The virtue of this approach is that it eschews spurious claims of truth and invalid inferences, each of which lead to spurious claims, outright error, falsehoods, dogma, propaganda, superstitions, overgeneralizations, the willful exclusion of certain truths --
I: I think a Positivist "core value" as you put it is that truth claims are justified by "proofs" -- not merely proofs of individual and specific claims as such -- but -- as true in virtue of being instances of universal laws -- certainly a fairly strong criterion for truth claims.
LP: Indeed -- but that is the point. The goal of LP is not to come up with a "plausible" view of reality, a convenience in making practical judgments -- but truth claims -- claims about the nature of reality. After all, to give an example, the Ptolemaic and Copernican -- the geocentric and heliocentric -- models of the solar system both work as a methodology for predicting the positions and motions of the planets. At most, only *one* of these models is true of the world -- the way things actually are.
I: You make a distinction between reality itself, and our perception of reality --
LP: Yes, claims about the former are true, or false. Claims about the latter are irrelevant; they are not knowledge.
I: David Hume and others have pointed out that general laws are universal statements: "Matter cannot be either created or destroyed", for example, applies to all matter, all the time.
LP: That is correct.
I: But Einsteinian physics has shown us that matter CAN be destroyed, converted into energy in fact.
LP: E = MC^2 in fact. Yes.
I: Isn't that a contradiction?
LP: Well, no, it turns out that the original claim that matter cannot be created or destroyed was false, at least for the sort of cases that Einstein addressed. It turns out that matter and energy are two different forms of the same thing, and in a special class of interactions, the law does not hold. So the generalization needs to be adjusted. Still, the original claim does hold for ordinary circumstances.
I: Well how should we generally interpret this "ordinary circumstances" approach? Can we say that under ordinary circumstance, light travels at 300 kilometers per second, but that under extraordinary circumstances it might go slower, or faster?
LP: Oh, no. The constancy of the velocity of light is an established law.
I: Established how? It seems to me that if you only observe light in "ordinary circumstances", then you might not be aware of what velocity it would have in -- as you said about matter -- in "extraordinary circumstances" -- under water, for example.
LP: Uh, no -- its velocity would be slower -- the constancy of the speed of light holds only in a vacuum --
I: An "ordinary" vacuum?
LP: Well, we speak of "standard conditions" --
I: Or sometimes "ideal conditions" --
LP: The point is, we do not merely make use of empirical observations, we use mathematical logic to handle the full set of cases. We cannot drop every apple from every tree to see if it falls to earth. Fortunately we are justified by mathematical induction to draw universal conclusions from well-defined specific cases --
I: Wait, wait -- how does mathematical induction justify claims about the physical world? And what is a "well-defined case"?
LP: First, experience with the natural, physical world shows us that -- miracles aside <he-he> -- the physical world is based on mathematical principles: 2 + 2 = 4 in practice as well as in theory. Second -- and this is very important -- in science, when we make generalizations from specific cases, we "abstract out" the universal features of a particle case. When we combine two moles of hydrogen with one of oxygen --
I: -- a "mole" being a basic quantitative unit of atoms of a particular element or molecules of a particular compound --
L: -- a pair of jacks, a six-pack of beer, a case of scotch (12), a gross of eggs (144), a mole of hydrogen atoms -- when we combine two "units" of hydrogen with one of oxygen, we get one "unit" of water -- H2O. We do not take into account the atypical properties of individual hydrogen atoms, say -- one might be a deuterium atom --
I: -- with a neutron added to the hydrogen nucleus --
LP: Yes. That would be an "accidental" property of that atom in this case. The law states that two moles of hydrogen atoms and one mole of oxygen atoms forms precisely one mole of the molecule H2O.
I: That reminds me of an experiment in Davis and Hirsch's book "The Mathematical Experience" in which they demonstrate empirically that 1 + 1 = 1.
LP: Wha --
I: They take -- and I have tried this -- one cup of air-popped popcorn to which they slowly pour in one cup of milk, allowing the popcorn to absorb the milk, The result is one cup of -- er --mush.
LP: Well -- that is --
I: -- not ordinary circumstances?
LP: Well, a cup of popcorn added to another cup of popcorn would make two cups --
I: So are we "mixing apples and oranges", to mix metaphors? Or "mixing hydrogen and oxygen"?
LP: Well, no -- you're playing with the standardized conditions --
I: But I guess my question is, what does Logical Positivism have to say about these "ordinary circumstances" and "standardized conditions". How, for example, do we judge when it is the case that we are properly abstracting the essential features of an object or event, and when we are merely over generalizing? I say this, because we do have scientifically conducted tests -- putatively scientific anyway -- which claim to measure "psi" forces (telepathy, for example), yet which are clearly not sound in their results. Now obviously, given that we have set up a set of procedures and guidelines (Logical Positivistic procedures and guidelines, for example) for what constitutes "knowledge", we can differentiate what -- according to the LP standard -- counts as "knowledge". My question is what justifies the LP standard?
LP: Its -- objectivity -- its universality --
I: Well, lets take a universal law -- a generally accepted law -- the law of gravity, which is fairly uncontroversial. We have the famous inspirational image of Newton watching an apple fall from a tree, and his Law of Gravity which states -- among other things that objects are gravitationally attracted to one another: release an apple, and it and the earth are attracted to one another -- the attraction -- the weight -- being on the side of the much larger earth.
LP: And I would point out that the witnessing of the falling apple does not constitute PROOF of the law, only a metaphor. Newton used empirical observation of planetary motion in a vacuum to construct a mathematical model incorporating only those general features common to all matter to form his law.
I: Yes, but how does this law deal with the apparent exceptions?
LP: Exceptions?
I: Yes. I hold a helium balloon. I release it. the balloon goes u--
LP: That is not an exception! The surrounding air is heavier -- subject to greater gravitational attraction -- than the balloon, which is pushed UP by the air. In a vacuum, the balloon would fall like a lead balloon--
I: Well what about a pigeon?
LP: A pigeon?
I: I let go of a pigeon, it flies SIDEWAYS. The pigeon is heaver than air, by the way --
LP: That is like the balloon -- gravitational forces imparted to the balloon and to the pigeon are superceded by other forces, the force of the surrounding air, or the force of the pigeon's wings upon the air.
I: And if I drop a hint?
LP: A HINT? I don't understand -- oh, "drop" -- you are playing with semantics now. "Drop" does not mean the same thing in the physical and the metaphorical cases. Puns are not knowledge -- they play on superficial similarities --
I: Aren't alchemy, and astrology and other pseudo-sciences based on those sorts of similarities? In alchemy, for examples, the chemical process of purification of an element occurs in a "crucible" -- the act of "crucifying" is analogous to Christian purification of the soul, and the transformation of -- lead, say -- into gold is a process of purification analogous to spiritual cleansing.
LP: Exactly! A theory of chemistry interaction that is modeled on the airy conjecture of spiritual meliorism! Lead is "redeemed" into gold. Only we know better through more rigorous investigation that lead is not "better" than gold, that atoms of one element do not "evolve" on any reasonable interpretation of the events, and that if anything, gold is not lead with the impurities removed, or improved -- and that the alchemic account is simply -- wrong.
I: The issue of the significance of "proof" would take us into many interesting areas -- but for now I would like to look at Probabilistic Inference -- the importance of which Locke discusses in his Essay on Understanding as I recall. As much as LPs exalt complete, sound proof for justification for its truth claims, the fact is, we cannot "prove" every truth claim that is made, can we? In actual practice, we do not justify -- logically prove -- every universality we then appeal to in support of some particular case. We see a lot of white swans, conclude that all swans are white on the basis of this, then -- and only then, once the "all swans are white" law gains public acceptance, the black swan of the family turns up, and we are forced either to make an embarrassing retraction, or to come up with a plausible "spin" on the now-discredited rule in order to save it, and Logical Positivism's principles and methodologies.
LP: You make it sound so shoddy and haphazard --
I: Well, isn't it? I am talking about actual implementation -- the *practice* of Logical Positivism in the "workplace" so to speak. It would seem to me that a theory of knowledge -- especially one as labor intensive and as exacting as Logical Positivism -- ought to have a good productivity record. As I mentioned earlier, Hume, and others, have claimed that we routinely do two things that are NOT justified by LP -- not even permitted by LP in fact.
LP: They are -- ?
I: Issuing claims of perceptions of Causal Interaction, and inferring universal statements from Inductive Generalization.
LP: Mathematical Induction, you mean.
I: It includes that, only -- it is not so mathematical -- and it is not all that hot among some mathematicians --
LP: The Finitists, you mean, those --
I: Well let's not get ahead of ourselves. Lets take a look at induction, "universalization" and probability inference, shall we?
LP: All right.
I: Generally speaking, knowledge based on true premises and valid inference takes the form of "If X, then Y"?
LP: That is correct for inferential knowledge, direct observation take the form of "Y is observed to be so".
I: So if I look at the bottom of my shoe and see a hole, then the claim "I have a hole in my shoe" is justified by direct observation. On the other hand, if I step into puddle of water and my foot feels wet, I make an inference, such as (1) my foot feels wet --
LP: Direct observation --
I: (2) If my foot feels wet (and it does), there is water in my shoe --
LP: Logical inference based on (1), and on a bit of general knowledge of the "conditional form" --
I: Well that is what I'm getting at: what justified (2) in this case? My foot could feel wet for other reasons -- my nerves are not working right, my foot is bleeding -- there are other possibilities --
LP: I see what you are getting at -- but you are misrepresenting the reasoning process -- oversimplifying it --
I: How so?
LP: You attach weights to likely inferentials. Look, you pick the "conditional" knowledge on what is likely and appropriate. (1) Your foot feels wet. (2) is really a conjunction of all the possible ways your foot could become -- or feel -- wet; your nerves are shot, you have a hole in your sole, it is raining inside your shoe, your foot is melting, you are bleeding, and so forth. Given the conjunction of "your foot is wet" and "you just stepped into a puddle", it is reasonable to pick the "if your foot feels wet, it is because there is water in your shoe", especially given the conjunction of "my foot has started to feel wet right now" and "at the very same time, I just stepped into a puddle of water".
I: You seem to be appealing to close conjunction (in this instance), and to the general rule of "constant conjunction". But as Hume has pointed out, the constant conjunction of two events is insufficient to PROVE connection, much less a causal relation between the two, much less which (if either) is the cause and which the effect. It may be for example that both are the effects of some third circumstance -- there are other possibilities --
LP: But let us be reasonable -- experience SHOWS us that --
I: Now wait a minute. Logical Positivism says nothing about our "experience" of events, much less any rationale for concluding from "experience" alone that such-and-such is the general rule. We OBSERVE A, and then B, and we observe A and then B *again*, and a third time, and a forth -- how often must one conclude from that that if A then B, or "A causes B", or "A is the harbinger of B", or whatever you wish to conclude. And where, in the name of Hume, do we observe the so-called "casual link" between A and B, no matter how many times we observe the two together?
LP: We can infer the link --
I: Based on what? Yes, you can argue that given "A and then B" we infer "A *therefore* B" -- but what is the justification for this conclusion?
LP: Well, one way to argue this is that if the two events or circumstances are related in some way -- one is the cause of the other, or both are the effect of some other common cause, or that there is some other connection -- then they are very likely to appear together, and not separately, and that one -- the cause -- is very likely to appear before the other -- the effect.
I: "Appearance" being our observation -- the first class of "knowables", the second being our inferred generalizations?
LP: Yes.
I: And "likelihood" -- the probability of A then B (or as another British empiricist and mathematician, Thomas Bayes put it, the probability of A, given that B is the case) -- is justified by -- what?
LP: I don't know what you mean.
I: Well, assuming that (1) your foot feels wet -- a direct observation -- then what justifies your claiming (2): if your foot feels wet, then there is water in your shoe rather than (2'): if your foot feels wet, you are experiencing a deteriorating nervous system (your brain, perhaps), or (2' '): your foot is bleeding, or (2' ' '): some ice from your drink slipped down the side of your sock and down into your shoe, and so forth. What is it that you are observing, or know from general truth claims (knowledge) that justifies in ultimately drawing the *sound* logical conclusion that water leaked in through a hole in your shoe and not as a piece of ice from your drink?
LP: Well [laughing] I can always look at my shoe --
I: But we can't always do that. Remember Einstein's image of the universe as a sealed pocket watch we can observe, and speculate upon, but cannot open up. But let's take a look at "probably inference", shall we? Suppose we have -- let's make it interesting -- a bottle of 499 orange-colored vitamin E capsules and one black-colored licorice jelly belly about the same size, shape and feel as the capsules.
LP: All right.
I: Now keeping in mind that we are making a metaphysical assumption -- that there ARE 499 orange vitamin E capsules and one black jelly belly candy in the opaque bottle, I wish to examine LP's reasoning in determining whether or not there is a jelly belly in the vitamin bottle, or that there are only vitamin E capsules in that jar.
LP: All right.
I: I jiggle the jar around for a bit, and draw an orange vitamin E capsule. I repeat this process twenty-nine more times, and come up with a total of thirty vitamin E capsules. What conclusion should I draw based on these results?
LP: Well, the rules for probability are not nearly as "chancy" as some take it to be. There is a well specified mathematical definition for what these results mean, which can be expressed in a couple of different ways. Basically, without knowing that there is a black pill in the bottle, given the number of "pills" in the bottle and the randomized drawing and the result, we can say that the chance that *all* of the pills are vitamin E is extremely high --
I: Despite that there is in fact a licorice jelly belly in the bottle?
LP: Yes. Probability talks about what is likely the case, not what IS the case. What IS the case is a matter of fact: there is a licorice jelly belly in the jar, or there is not. If we need to know whether this is the case, we need to go through the entire bottle. This is usually not the case in scientific matters; the natural world tends to "bunch up into discernable kinds". One classic example is basic chemistry, in which atoms of particular elements appear in molecules in reasonably balanced proportion. Sucrose is C12-H22-O11 while salt is Na-Cl and sulfuric acid H2-S-O4. You just don't find molecules with ten million hydrogen atoms and one carbon atoms.
I: But its that so because thee are no such molecules, or because we do not look for them, and hence do not find them.
LP: <laughing> Well, if there were such molecules, then there would be substances made up of these molecules, and we would examine such substances, and discover molecules that violated the basic principle of stoichiometry --
I: The principle that atoms appear in molecules in definite, discrete -- and relatively small proportions, as in water (H2-O) and not in varying amounts such as H3-O, H2-O2 and the like, or in proportions of a million to one --
LP: That is correct. I should add that all of nature seems to "bunch up", or occur in repeatable, normalized proportions. Cows tend to be all about the same size and shape and alike in other characteristics, as are pine trees, and diamonds and asteroids. There is no continuous variation between objects that are more cow like and less asteroid-like at one end of the spectrum, and more asteroid-like and less cow like at the other. If there were then it would be impossible to recognize, or view, much less make generalized claims of knowledge about them, to be sure. We could not conduct repeatable experiments, or make predictions about the nature or behavior of the natural objects of the world -- there would BE no science.
I: Then if there were only "one" of everything, even if -- *especially* if -- there were a great many one-of-a-kind objects in the physical world -- then we would not have "knowledge" of the kind you described earlier.
LP: Absolutely not. There would BE all of these infinitely diverse objects -- a billion different elements, sextillions of different molecules, sheer astronomical numbers of different *species* of life in a single backyard -- if you could even call them "species". Even if there were some relation between a parent and offspring of some life form, how would we know, unless we were present at their birth? We would know *nothing* about the relations between -- well -- anything. Causality, and kind, and properties would be completely invisible to us, just as in mathematics where irrational and transcendental numbers are impossible to write down completely, or compute with precisely, except when symbolized, as in the case of the square root of two, or "pi", or "e", or for any of the omega numbers --
I: Omega numbers?
LP: Infinitely long decimal numbers for which there is no short-hand way of denoting them. They are not the result of solving an equation, or summing up a series -- they simply have to be written out to be distinguished from each other -- and they cannot be completely written out; any process for writing them out, or for reading them, or computing with them would take forever.
I: That leads to another line of questioning. There were two serious blows against Logical Positivism in the early 1900s -- one was the Heisenberg Principle in physics, and the other was Godel's Incompleteness Theorem --
LP: They were -- they exposed limitations upon the Logical Positivist programme, I'd prefer to say --
I: I'll get back to Heisenberg in a moment. First, as part of the LP "programme" as you call it, Godel showed that you could not -- even in mathematics -- even in elementary school arithmetic, fully resolve the question as to which truth claims in arithmetic were true, and which ones were false -- despite the fact that every one of these statements IS either true or false.
LP: Well, that is because we set a higher standard of "proof" for the truth of a statement than others do . . .
I: It isn't enough to SAY that two to the billionth power is such-and-such -- you require a "proof" in order to claim that one *knows* the value of that expression -- can you explain that for me?
LP: Well let's take a simpler case by way of illustration. Let us show that 2 + 2 = 4 is a provable theorem of arithmetic --
I: What does that mean -- "a provable theorem"?
LP: It means that we not only assert that "2 + 2 = 4", and that it has a certain specific interpretation, but that it has a step-by-step proof, starting with our assumptions, and using logical inferences to reach our theorem. The assumptions we use for theorems of arithmetic are the five Peano axioms -- very basic axioms:
(Axiom 1) Zero is a number
(Axiom 2) If N is a number, then N ' (the successor to N) is a number.
(Axiom 3) Zero is not the successor of any number.
(Axiom 4) If the successors of two numbers are equal, then those two numbers are themselves equal.
(Axiom 5) (Induction Axiom) The set of natural numbers is defined as the set of numbers including zero and the successor of every number in the set of natural numbers.
I: What does that get us?
LP: It gets us the proof that "2 + 2 = 4". Look, we know -- it is given -- that zero is a natural number, right? Then the successor to zero is 1 -- a natural number by axiom (2), and therefore the successor to 1 is 2, also a natural number. Then we define addition as a successor operation: N --> N ' is defined as "addition by one": N + 1 = N '. From this we can show that 2 is zero plus 2 [Z ' ' = Z + 1 + 1 = 2], and then that 4 is 2 plus 2 more [2 = Z ' ' = Z + 1 + 1, and (Z + 1 + 1) + 1 + 1 = (Z ' ' ) ' ' = 2 ' ' = 3 ' = 4]. QED.
I: If Axioms 1 - 5 are indeed axioms, then what justifies their truth?
LP: The obviousness of their truth. Suppose zero was not a number -- what then? Besides, these axioms are *useful* because one can prove all of the theorems of arithmetic from them witout the need for any additional axioms -- the entirety of arithmetic is condensed in these five little statements.
I: A very neat "reduction" -- which is a hallmark of LP. But I am still concerned about these axioms and their applications. You say that the justification for the claim that zero is a number is that it is "obviously true"?
LP: Well, we have our observations from nature and from working with numbers --
I: So mathematics is *empirical*? What about the cup of popcorn added to the cup of milk --
LP: That's not math -- that's cookery. Look, some axioms, like (1) are definitions -- arbitrary and useful for proofs -- others are choices -- like Euclid's Fifth Postulate -- you know: "parallel lines never meet". The fifth postulate was really a choice made between three possibilities: parallel lines never meet, parallel lines converge, and parallel lines diverge -- three geometries, three choices. Or if you like, we can take a broader view of geometry and drop the fifth postulate altogether: Projective Geometry, in which the distinctions between "lines" and "points" are dropped --
I: That reminds me -- axiom (5) -- that all of the successors -- succeeding successors I mean -- of zero are natural numbers. Couldn't you prove that by applying axioms 1- 4?
LP: A finitist might insist upon it -- Finitist mathematicians doubt the metaphysical existence of infinity -- they regard it only as a "device". If you were to go about "proving" axiom (5), you would start with the first four axioms, and begin by proving that zero is a natural number, which is can be done in one step: invoke axiom (1). We prove that 1 is a natural number by applying axiom (2): showing that it is the successor of a number we know to be a natural number, namely, the number zero. Now that zero *and* one are shown to be natural numbers, by repeatedly applying axiom (2) to the result, w can eventually show that any given positive integer 'n' is a natural number by showing it to be the nth successor to zero --
I: So, you've proved the fifth axiom --
LP: No -- we have only proven that all of the integers up to some finite number 'n' are natural numbers. There are infinitely many more natural numbers, so it would take an infinitely long proof to show that *all* of them are indeed natural numbers.
I: Wait a minute -- you mean it is TRUE that all of the numbers zero, one, two, three and so on are natural numbers -- you just can't PROVE it?
LP: [Sounding embarrassed] Eh -- no. The problem is that the proof is not finite. If you prove that *all* of the number of a given set up to 'n' are positive integers -- natural numbers -- *that* doesn't prove that the next number in your set (n + 1) will be a natural number -- not until you construct a proof for that number as well -- and then you still have to do that for all of the succeeding numbers. you can't just say "this is the set of natural numbers because the first one million of them are natural numbers -- ". You either argue in a circle: this set of numbers is the set of natural numbers because we say it is, or you accept it as an axiom --
I: Which is a formalized way of saying exactly the same thing. Now about Godel's Incompleteness theorem --
LP: Which in a non-fancy way of stating is the fact that when you set up a mathematical language, with symbols and quantities and operations and the like you are, in virtue of the grammar of the language, able to form "statements", like "2 + 2 = 4", "7 - 7 = 12" --
I: Which is *false* --
LP: Of course. the problem, as Godel discovered -- proved -- is that when you set up any mathematical language and start grinding out statements, you immediately face the "characterization problem" --
I: Which is the question: "Does this particular statement go in the "true" bag, or the "false" bag?"
LP: Exactly. It's easy at first: "2 + 2 = 4" is true, "7 - 7 = 12" is false -- but when you get to general statements, like "x + x = 2x", and "There is no number 'n' that satisfies Fermat's Last Theorem", the strategy for finding a proof gets harder and harder -- and longer and longer. In the case of universals, you are dealing with proofs that are not merely infinitely long, they are longer than that -- the language cannot handle such statements -- or the proofs that are required to show they are true (or false). Of course we can always do as we did with Euclid's Fifth Postulate: list the alternatives and pick one of these as a new axiom --
I: But which one -- and *why*! It sounds to me like you are ad hocking a system of knowledge, not establishing provable "truths".
LP: well, yes, that *is* a problem --
I: I wish to move on to physics now -- the other LP darling. The LP claim is that physics is "mathematics actualized".
LP: Well, some of my colleagues used to believe that --
I: Burned by "The Uncertainty Principle"?
LP: Well, the thing about that is we started out -- 1890, 1900 -- with a Newtonian Universe based on Euclidian Mathematics, and by the 1930s we were working with a post-Einsteinian Universe based on Godel's Incomplete Arithmetic. It kicked the notion of "formal proof" out from under us.
I: How did the Heisenberg Uncertainty Principle hurt?
LP: Well, even if we could never measure the physical universe in all of its properties and states and form and structure in *practice*, we could at least do so in theory, and chalk up the error to measurement flaws. But quantum mechanics -- the new physics -- it is like a carpenter discovering that however precise a ruler he has, he can never measure his tables, chairs, sideboards any more accurately that one inch give or take. no matter what he does, he cannot make any measurement, cut or sand any piece of wood, drill any hole -- more accurately than to within one *inch*.
I: Quantum physicists are split as to whether this "margin for error" is the result of the ultimate limit of our measuring tools, or whether objects *themselves* -- tables, chairs, slats, holes -- are "fuzzy" in dimension. Electrons are not hard little balls, but probability manifolds. A certain electron is said to be "mostly within this volume of space", and not definitely here or there. Is the fuzziness of our descriptions -- our *knowledge* -- of physical reality due to the limitations of our instruments, OR, to the definiteness of the objects of our world?
LP: Answer that definitively and then go pick up your Noble Prize in Oslo come next year [laughter].
I: Putting aside the perceived fuzziness of the physical world it seems to me that there are other, perhaps more immediately series problems with LP physics --
LP: You are figuring this out, *now* [laughter].
I: Well, going back before the contemporary physics of the last century, back to the Empiricists, in fact, of the 1700s -- many skeptics -- Berkeley and Hume, to note only a few --
LP: You are talking about the problem of "causality" I presume?
I: Yes, as Hume put it, we see the cue ball, we see the target ball, the cue stick, the hole and the felt pool table top -- but we do NOT see the "causes" when the cue and the balls move. we merely conjecture them and hope for the best.
LP: Well, I use to say that "probability" was our way out -- that if something happened 99 times in the same circumstances, then it was safe to bet it would happen the 100th time around -- but purists still hold to that, while cynics tend to hold to the opposite -- you know the guys who kept betting on black at Monte Carlo when the ball landed on red 22 times in a row -- that's about one in four million --
I: -- and the bank broke the gamblers. Doesn't that apply in the natural world as well -- the mathematical world as well, too. The highly useful Pythagorean theorem is not only provable, it is easily generalized to any number of dimensions. The four color map theorem, the Goldbach conjecture, Riemann's Hypothesis, and Fermat's Last Theorem, however, have been terribly hard to solve -- Riemann and Goldbach are still unsolved as I understand. True, they may not have the immediate practical applicability of Pythagoras's theorem, but --
LP: You will note that the theorems you just mentioned are mainly "pure math" theorems -- they seem to be the most difficult ones - the "fun and games" problems. The practical ones we can knock off fairly easily -- why do you suppose that is?
I: An interesting question -- well, for one, the physical world -- insofar as it is mathematicizable -- seems to be much simpler than the more varied and extended world of mathematical objects, most of which have *no* analog to reality --
LP: Infinity, for example -- "primeness", "spaces" -- a lot of them I suppose -- though Newton's "Three-Body problem" is a very practical one in orbital mechanics, and yet it has eluded exact solution.
I: Getting back to physics and probability -- and my point about -- "long-shots" I guess you'd call them -- if our knowledge, about mathematical or about physical entities is merely a matter of betting on the short-odds -- on the easily provable "laws" --then isn't our knowledge of the world -- the mathematical or physical world -- a matter of "cherry picking" -- knowledge is whatever is "easy for us to prove" by some fairly objective methodology of proof?
LP: You put that rather grimly, but in practice, yes, our methods for ascertaining knowledge in the "hard" sciences is skewed towards the easily demonstrated cases. Which leaves open the question as to what are the *important*, the fundamental -- axiomatic or "law-like" generalizations, laws, rules or what have you. That the general rules we have discovered about the mathematical and physical world tend to be a function of their ease of discoverability rather than of their fundamental importance or "support" in the structure of knowledge -- that is troublesome. It makes you wonder what we don't know, and why we do not know it --
I: And whether it is knowable at all --
LP: Indeed. I am a little less certain of the claims of LP epistemology than I first thought I would be. One final topic for this time -- and then I would like to talk to you again about a separate set of matters, namely "persons" and their properties in the LP model of reality -- and that is this issue of causality. LP claims to identify, isolate, objectify and structure what is basically real -- the stuff of hardcore reality, apart from religious bias, from rationalistic notions having no support in reality -- LP has --and still is, for many people -- been a epistemic purgative. Anything that does not fit, that does not have actual support in nature (or in mathematics), that does not have *some* necessary and visible basis in observed phenomena -- is suspect. Hence, no Santa Claus, no "crystalline spheres", no "best-of-all-possible-worlds", no "invisible" or otherwise unmanifest phenomena --
LP: "Hidden forces", as Einstein called them. He was a Classical Newtonian to the bitter end.
I: -- and yet, ever since Hume, we have confronted, and never really nailed down -- one of the most fundamental, relied upon and trusted in of LP concepts: causality. Now in math, as complex and intricate as it is, the "laws" tend to lie on the surface -- the law of cosines, for example, the rules for mathematical operations, and so forth -- they are all justified by explicit, well-defined "rules". But -- physical causality -- laws of nature -- Hume disparagingly called them "constant conjunctions"; individual cases observed again and again and again -- like the finite instances of your infinitely long mathematical "proofs" -- the ones that are never finished for the general case. You take probabilistic samples, and deduce a Law from then on the basis of the absence of counterexamples. Is his -- *knowledge*?
LP: Well, it is certainly better than guessing, faith, or out-of-the-blue ideas deduced purely from pet views about the way things ought to be.
I: But even so, the more one examines the philosophy of science, and realizes the holes in major -- even structural -- inferences about the world -- it -- is disturbing to say the least. One of the things that bothers me the most is the problem related to Bayes theorem of probability: the probability that the cause of Y was X, given that Y *did* occur. We do not *perceive* cause -- observation being the first principle of your logical positivist system -- and we cannot *infer* a cause from an insufficient number of -- well -- insufficiently *causally* related cases -- valid inference being your second principle. I think we have run out of LP principles. But I would like to question you on the LP position on very troublesome, though not at all problematic entities: people, minds and related objects.
LP: I'll try to give you the LP position on these -- but I assure you that you won't be happy . . .
I: The question is whether my unhappiness is a fault in myself, or in Logical Positivism . . .