01.19.10
Posted in Relativity, philosophy, physics, science at 12:22 am by nogre
[This is something I wrote before I had this blog, but I really like it and hope the readers here will find it interesting.]
The task of explaining Aristotle’s theory of place lies in the interpretation of this sentence: “Hence the place of a thing is the innermost motionless boundary of what contains it,” (Physics IV 212a20). Now the idea of a motionless boundary for perceptible and obviously movable objects seems impossibly counterintuitive. However, using Aristotle’s comments into the nature of place, we can understand how this theory extends beyond a simple boundary theory and into the modern era.
His discussion is started by making an observation:
The existence of place is held to be obvious from the fact of mutual replacement. Where water now is, there in turn, when water has gone out as from a vessel, air is present; and at another time another body occupies this same place. The place is thought to be different from all other bodies which come to be in it and replace one another. What now contains air formerly contained water, so that clearly the place or space into which and out of which they passed was something different from both. (Physics IV 208b1-8)
Aristotle then makes some tentative moves into defining what could possibly fulfill the role of place. First he discounts any idea that place could “be body; for if it were” he says, “there would be two bodies in the same place,” (209a6) and shows how this causes untold amounts of theoretical difficulty (ibid.). Then he discusses how, as that which “primarily contains each body” (209b1), place could be viewed as the form or the matter. Again he discounts either of these possibilities by noting that neither of them can be separated from a thing, whereas place may be.
The analysis turns at this point asking, “How many ways one thing is said to be in another,” (210a14) in the hopes of landing upon a useful interpretation of the notion of being-in. Aristotle entertains the idea that a thing may be in itself, or more specifically, in itself “qua itself or qua something else” (210a27). He gives the example of a jar of wine being in itself in virtue of the whole’s description in terms of its parts: the jar of wine is not reducible to a jar or some wine, but to their specific combination. Hence that which is a subject may potentially be a container as well, as the jar is the container of the subject ‘jar of wine’. However, he says this is impossible as no object is actually like this: the wine would have to be an equal part container, and the jar an equal part wine, else the whole of the ‘jar of wine’ will not be completely in itself. It is in virtue of being different that the jar and the wine may come together, and hence he concludes that, “since the vessel is no part of what is in it (what contains something primarily is different from what is contained), place could not be either the matter or the form of the thing contained,” (210b27).
Aristotle then considers two sorts of boundary theories. First, place is such that it is “some sort of extension between the extremities,” (211b7). However, this sort of boundary exists independently of what is bounded and is permanent. Insofar as anything moves, the place will change, and there will be two problems generated: 1. The boundary between, for example, the wine and air moving in a jar will be exactly coincident with the boundary between the air and wine, and each of these two boundary extensions will be partially coincident but must also be unique, and 2. There will be a place at the boundary of the displaced place, and so an infinite regress of places associated with previous places will be generated.
Finally Aristotle says, “place necessarily is… the boundary of the containing body at which it is in contact with the contained body,” (212a6). Then, interestingly, he says, “Place… is rather what is motionless,” (212a17) and then, “Hence the place of a thing is the innermost motionless boundary of what contains it,” (212a20). So how are we to make sense of containers that give us the boundaries but also do not move?
If we take our interpretation directly from the previous discussion of boundary, Aristotle seems to have made an awfully strange claim: place is a like a container that something perfectly fits in, and yet that container cannot itself move although what is contained therein necessarily can. However, I would like to suggest taking him at his word in regards to place being necessarily a boundary. It is necessarily a boundary of the containing body which is in contact with the contained body, but it is not sufficient that it be the boundary that most have in mind. What has been missed by regarding the boundary as necessary and sufficient is the incorporation of Aristotle’s primary intuition into how we know place exists.
For us to notice any motion we need something stationary relative to the thing moving such that we may observe the motion as motion. A stationary backdrop is necessary to view change. Consider this example: it is a common experience to be sitting in traffic next to a bus. When the bus starts to slowly pull forward sometimes it is possible to get the sensation that you have started to move backwards. This sensation lasts until some other fact informs you that it was not you who were retreating, but the bus advancing. In the first instance, the place that you inhabited was defined in terms of a stationary bus: any motion that occurred was relative to that fixed point. When you realized it was the bus, and not you, that was moving, whatever it was that informed you that you were stationary became incorporated into the place. Perhaps you saw a building and since it was not moving your senses of place and motion were reevaluated to accommodate this new information. Hence two different places were involved in this scenario: one corresponds to your (belief of) moving backwards and the other to the forward motion of the bus. The first place has the innermost motionless boundary [illegitimately] defined in terms of the bus [since the bus was in motion], and the second has an expanded sense of place to include something motionless relative to both you and the bus. Place, therefore, not only encapsulates that which is moving, but also whatever is observing the motion and an independently stationary object.
When Aristotle says, “First then we must understand that place would not have been inquired into, if there had not been motion with respect to place,” (211a12) he was not making an idle comment on the ‘discovery’ of the phenomenon of place. Instead, he was beginning his analysis with the most important feature to be explained. Galilean relativity holds fundamentally that motion can only be defined relative to some “system of coordinates” (Einstein p. 14), i.e. something motionless with respect to the moving object. Although Aristotle did not have the benefit of Descartes’ mathematical works he still recognized the need for a reference system unique to each motion.
Now that the meaning of the ‘motionless’ criterion has been explained, what does ‘innermost’ mean if the boundary may include objects at a significant distance from the one we are describing? The Earth is, in some sense, the place of all sublunary objects, but it is surely not the innermost boundary. Again appealing to the exact phenomenon that Aristotle was dealing with yields the correct explanation: the motion of an object with respect to its place cannot be described by saying that the place of the thing is Earth. Instead the innermost boundary should be the first boundary for which the description of motion and place of the thing make sense. In the example of swirling wine in a jar (with the jar held steady but the wine still moving), the place of the wine can be said to be the jar because it describes the closest motionless boundary. One might say that the place of the wine is equally on the table, or in someone’s hand, but both of these descriptions are related to motion other than that of wine swirling in the jar, and are really just shorthand for ‘the jar of wine is on the table, or in hand’ (consider, “The wine is on the floor,”: no swirling to be had). The correct place of the moving bus example was the street, which could reasonably include things such as houses and other stationary objects, the most important one being the road. Aristotle gives the ancient version of this, just before his final definition of place, in terms of the motion of a boat, such that it must be defined in terms of the whole river, which is taken as stationary (212a19).
One may think that I am being too charitable to Aristotle at this point because it may look that I have implied that the ‘innermost’ criterion of the theory will do all the work of translation between coordinate systems, which it does not seem to do. However, Aristotle allows for the fact that we may treat a vessel such as a boat as a “transportable place” (212a14). Hence we may allow for the things within a boat to have motion independent of a preferred fixed reference system. This connection drawn between vessel and place gives the final aspect needed for a true relativity theory. Historically the example of a boat on a river (212a19) is striking because Galileo uses the same example of a ship in his Dialogue Concerning the Two Chief World Systems (Salgado).
Commentary
Historically Aristotle’s theory of place has been treated as a simple boundary theory coupled with a few very strange statements that were assumed to be preliminary investigations into different methods of solving blatant paradoxes (King 91). My suspicion for why this happened is twofold. First, philosophy of place sounds like the most dry and uninteresting subject possible and hence it was not given its due study time over the course of history (after Aristotle that is). Secondly, the modern formulation of relativity is given according to Einstein’s theory of Special Relativity. Insofar as this is a theory of relative motion, it is easy to overlook the fact that it applies to objects without motion, things in place. Aristotle’s formulation is none the weaker because of the way he cast his theory, but it is much more obscure to the modern reader, as Aristotle’s relativity is developed in a somewhat inverse way with respect to modern relativity.
Bibliography
- Barnes, J. ed. The Complete Works of Aristotle v.2, Princeton U. Press, Princeton 1995
- Einstein, Albert Relativity: The Special and General Theory Lawson trans. Pi press, NY, NY 2005
- King, H. R., “Aristotle’s Theory of TOPOS,” Classical Quarterly #44 1950, pp. 76-96
- Salgado, Rob “Galileo’s Parable of the Ship” http://physics.syr.edu/courses/modules/LIGHTCONE/galileo.html Accessed 2/24/06
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12.31.09
Posted in Relativity, biology, evolution, measurement, philosophy, physics, science at 4:48 pm by nogre
I have previously argued that the history of species must be treated like a evolutionary trajectory: we can only appreciate a species in a relative sense, just as we must evaluate physical trajectories relative to our own motion.
But what happens when we try to measure the very small in physics? We find there is a limit to the precision at which we can measure, as given by the uncertainty principle.
This suggests that there may be some similar limit when it comes to measuring small changes in species. The more we try to pin down exactly what a species is, the less sure we will be about its future and the more we measure the direction the species is heading, the less sure we will be about exactly what constitutes that species.
If genetic drift is just another way of saying that we cannot pin down the exact genetic make-up of a species then drift may be considered to be an instance of the uncertainty principle.
and HAPPY NEW YEAR
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04.24.09
Posted in Relativity, biology, evolution, fitness, philosophy, science, technology at 4:48 pm by nogre
I’m sure the dearth of posts here has not gone completely unnoticed. So what in the world have I been up to? While it is possible that I had decided to forgo my normal practice of just making up philosophy as I see fit, this, of course, is ridiculous.
I do not know if any of my readers have been around since the beginning, but this blog started out with a good deal of my philosophy of biology. The one-sentence description of my philosophy of biology is: I have relativized the theory of fitness within evolution and comported the rest of evolutionary theory to make it work.
In the last few months I’ve been teaching myself to program and, as of today, I have incorporated as much of my theory of evolution into a simulation as I can possibly hope to accomplish (at the moment). All that is left is to get all the bugs out.
It is not every day that I can say that I have created something that is a direct result of a theory of philosophy, moreover a theory that I have personally developed. I’m pretty stoked. There is still a good amount of work getting the thing to actually run from this point forward, but at least none of the issues will be theoretical, just technical.
And of course there is no guarantee that things will go the way I want them to, but at least I gave it a shot.
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02.22.09
Posted in Relativity, measurement, ontology, philosophy, physics, science, time at 11:17 pm by nogre
Measurement takes time; measurement is a process. So the measurement of time immediately yields this theoretical issue:
Since measurement takes time, our ability to break time into ever smaller pieces will always be proportional to the method of measurement used. The faster our measurement device that measures time, the more divisible time will be. Insofar as there are limits to how fast a measurement process can occur (relativistic or other), there will be limits on the lengths of time we can measure. From this perspective, time is discontinuous: there will be a point at which we can no longer split time into smaller pieces.
From a different perspective, time must be continuous: we can start our measurement of time whenever. Since there are no restrictions on when our measurement may begin, each and every instant must be just as good as every other instant, hence time is continuous.
So which is it: Is time continuous or discontinuous?
Or is the question badly formed? The discontinuity argument is based upon the ideas of measurement and relativity. The latter argument, for continuity, is based upon what might be considered a fact of modal reality. Perhaps the two arguments are not talking about the same thing.
I can’t give an end-all be-all answer to the questions of time, but here is my opinion: Time is continuous, but when we start to do scientific activities, time can and will only be able to be measured discretely. Therefore the two arguments are not using one word to describe two different phenomena.
The question then becomes how doing science limits what we can observe.
This might sound like an extremely unlikely situation, but consider the case of organized sports. When playing a sport or game you are bound, restricted, to following certain rules. However, by following these rules, you and the other players can demonstrate skills and abilities that you otherwise would not have been able to observe: Lots of people may be in shape, but only a small fraction of those people are professional athletes. Those athlete demonstrate their superior physical and mental prowess by performing on the game field by being restricted by the official rules.
Getting back to science, does it now seem so unlikely that we restrict ourselves in certain ways in order to accomplish other tasks? For time to be scientifically useful, we need to have some sort process that has a fixed point from which to start counting from, and a unit to count. Then we can compare an unknown process to this known process, and we have done so with much success.
This comparison could not have occurred without the introduction of an arbitrary fixed point and unit of measurement: by restricting our concept of time to these particular processes we enable ourselves to perform scientific research. Research is not possible if we use the unrestricted modal notion: no comparison can be made because there is no inter-modal process to compare a worldly (intra-modal) phenomenon to. But with the use of fixed points, units and processes, we also become subject to relativistic limitations. It seems like a very small price to pay considering the success of science.
To sum up: time is subject to modal considerations, which gives it special properties such as being continuous. Once we start to do science, though, we restrict ourselves to the non-modal aspects of time, which allows us to use it as a tool in scientific research. This also makes time appear to have different properties, but upon closer study, these properties are artifacts of the measurement process and not time itself.
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01.01.09
Posted in Relativity, economics, philosophy, science at 10:59 pm by nogre
Readers of this blog may have noticed a lack of updates recently. I can’t apologize: I’ve been eating, breathing and drinking philosophy for so long, that now that I have written everything I wanted to write, I feel free. I wish it on all of you. [Happy New Year Everyone!]
But this doesn’t stop me from thinking. I was at a Christmas party and got talking with an Indonesian economics grad student. He was researching economic methods Indonesia could use to become treated as a major world power, such as investing in ports with international business significance. Interesting stuff.
Unfortunately my knowledge of economics is woeful. However, when I pressed him to explain exactly how his economics works, he used ethical terms. This gave me the idea that economics is fundamentally based in ethics. When economists speak of value, this use of value is not different than the value we use in ethics, only a bit more abstracted.
Money used to represent a commitment of the issuing institution to having a certain amount of a precious metal on hand. The cash was a proxy for that metal. Metal, of course, has no inherent value: it is just a lump of metal. What gives a lump of metal value is its properties that people use for specific purposes, and these purposes are fulfilling the commitments we have in daily life. Hence money is an eventual proxy for commitments.
Now, commitments are a relativistic metaphysical substance, something I have much more experience with (man that’s a funny thing to say). Relativistic metaphysical substances can be analyzed along the general guidelines of physical relativity: there is a general theory, there is a special theory, and then there is quantum.
The forces of macroeconomics can be conceptually aligned with a general relativistic theory, and microeconomics with special relativity. Unfortunately everyone already understands these things so there is little hope of finding some inefficiency to exploit.
What people don’t get is quantum mechanics: it is just accepted that things are weird in the quantum world. My view is that quantum phenomena are a highly sophisticated relativistic measurement issue (yes, I have seen all the data against this view and I am still convinced). This allows me to look in the world of economics for similar relations and, lo and behold, impulse buying fits the schema.
Impulse buying appears to unbelievably unstudied: it has 4 whole paragraphs dedicated to it in Wikipedia. I know this isn’t the best judge of research, but other economic topics in Wikipedia seem to have textbooks written about them and impulse buying has 1 academic journal reference (about how distraction affects brand selection during an impulse buy, not exactly the underlying theory) and a reference to a ‘Natural Parenting’ magazine article. I feel like I can declare myself an expert right now: I am the foremost leader in the economic theory of impulse buying.
I guess now that I am done with philosophy I can start a business to see if my theories are correct…. never thought I’d end up in experimental philosophy, but it just goes to show that you can never say never.
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10.18.08
Posted in Relativity, measurement, science, time at 9:32 pm by nogre
There is something about time that I can’t seem to stop thinking about.
We measure time by agreeing upon an event and then counting from that point onward. Today is October 17, 2008 AD. It is this AD that keeps my attention. It has been 2008 years, ten months and seventeen days since the birth of Jesus of Nazareth: AD stands for Anno Domini, or year of our lord. Those not wanting to be explicitly Christian use CE, which stands for Common Era, which is just a nice way of saying the same thing without recognizing Jesus as the lord. Wikipedia dates the use of this term to 525 AD, though this is how everyone has been measuring time forever. AD began to be used in 525, but before that people just used other events (like natural disasters, battles worn or lost, etc.) as starting points to count the date from.
The only result is that time is not universal but relative to whenever people agree to start counting from. This is nothing new, but maybe like The Ring, if I pass it along, then it won’t bother me any more. If you become similarly afflicted, I apologize, but you know what to do.
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08.18.08
Posted in Relativity, Special Relativity, biology, epistemology, evolution, fitness, independence friendly logic, logic, measurement, mind, philosophy, physics, science, technology at 1:26 pm by nogre
I just returned from a cruise to Alaska. It is a wonderful, beautiful place. I zip-lined in a rain forest canopy, hiked above a glacier, kayaked coastal Canada and was pulled by sled-dogs. Anywho, as on many cruises, there was a casino, which is an excellent excuse for me to discuss probability.
What is probability and where does it come from? Definitions are easy enough to find. Google returns:
a measure of how likely it is that some event will occur; a number expressing the ratio of favorable cases to the whole number of cases possible …
So it’s a measure of likelihood. What’s likelihood? Google returns:
The probability of a specified outcome.
Awesome. So ‘probability as likelihood’ is non-explanatory. What about this ‘ratio of favorable cases to the whole number of cases possible’? I’m pretty wary about the word favorable. Let’s modify this definition to read:
a number expressing the ratio of certain cases to the whole number of cases possible.
Nor do I like ‘a number expressing…’ This refers to a particular probability, not probability at large, so let’s go back to using ‘measure’:
a measure of certain cases to the whole number of cases possible.
We need to be a bit more explicit about what we are measuring:
a measure of the frequency of certain cases to the whole number of cases possible.
OK. I think this isn’t that bad. When we flip a fair coin the probability is the frequency of landing on heads compared to the total cases possible, heads + tails, so 1 out of 2. Pretty good.
But notice the addition of the word fair. Where did it come from, what’s it doing there? Something is said to be fair if that thing shows no favoritism to any person or process. In terms of things that act randomly, this means that the thing acts in a consistently random way. Being consistently random means it is always random, not sometimes random and other times not random. This means that fairness has to do with the distribution of the instances of the cases we are studying. What governs this distribution?
In the case of of a coin, the shape of the coin and the conditions under which it is measured make all the difference in the distribution of heads and tails. The two sides, heads and tails, must be distinguishable, but the coin must be flipped in a way such that no one can know which side will land facing up. The shape of the coin, even with uniform mass distribution, cannot preclude this previous condition. Therefore the source of probability is the interdependence of physical conditions (shape and motion of the coin) and an epistemic notion (independence of knowledge of which side will land up). When the physical conditions and our knowledge of the conditions are dependent upon each other then the situation becomes probabilistic because the conditions preclude our knowing the exact outcome of the situation.
It is now time to recall that people cheat at gambling all the time. A trio of people in March 2004 used a computer and lasers to successfully predict the decaying orbit of a ball spinning on a roulette wheel (and walked out with £1.3 million). This indicates that after a certain point it is possible to predict the outcome of a coin flipping or a roulette ball spinning, so the dependence mentioned above is eventually broken. However this is only possible once the coin is flipping or the roulette ball is rolling, not before the person releases the roulette ball or flips the coin.
With the suggestion that it is the person that determines the outcome we can expand the physical-epistemic dependence to an physical-epistemic-performative one. If I know that I, nor anyone else, can predict the outcome until after I perform a task, then the knowledge of the outcome is dependent upon how I perform that task.
This makes sense because magicians and scam artists train themselves to be able to perform tasks like shuffling and dealing cards in ways that most of us think is random but are not. The rest of us believe that there is a dependence between the physical setup and the outcome that precludes knowing the results, but this is merely an illusion that is exploited.
What about instances in which special training or equipment is unavailable; can we guarantee everyone’s ability to measure the thing in question to be equal? We can: light. Anyone who can see at all sees light that is indistinguishable from the light everyone else sees: it has no haecceity.
This lack of distinguishability, lack of haecceity (thisness), is not merely a property of the photon but a physical characteristic of humans. We have no biology that can distinguish one photon from another of equivalent wavelength. To distinguish something we have to use a smaller feature of the thing to tell it apart from its compatriots. Since we cannot see anything smaller, this is impossible. Nor is there a technology that we could use to augment our abilities: for us to have a technology that would see something smaller than a photon would require us to know that the technology interacted at a deeper level with reality than photons do. But we cannot know that because we are physically limited to using the photon as our minimal measurement device. The act of sight is foundational: we cannot see anything smaller than a photon nor can anything smaller exist in our world.
The way we perceive photons will always be inherently distributed because of this too. We cannot uniquely identify a single photon, and hence we can’t come back and measure the properties of a photon we have previously studied. Therefore the best we will be able to accomplish when studying photons is to measure a group of photons and use a distribution of their properties, making photons inherently probabilistic. Since the act of seeing light is a biological feature of humans, we all have equal epistemological footing in this instance. This means that the epistemic dependence mentioned above can be ignored because it adds nothing to the current discussion. Therefore we can eliminate the epistemic notion from our above dependence, reducing it to a physical-performative interdependence.
Since it is a historical/ evolutionary accident that the photon is the smallest object we can perceive, the photon really is not fundamental to this discussion. Therefore, the interdependence of the physical properties of the smallest things we can perceive and our inherent inability to tell them apart is a source of probability in nature.
This is a source of natural randomness as well: once we know the probability of some property that we cannot measure directly, the lack of haecceity means that we will not be able to predict when we will measure an individual with said property. Therefore the order in which we measure the property will inherently be random. [Assume the contradiction: the order in which we measure the property is not random, but follows some pattern. Then there exists some underlying structure that governs the appearance of the property. However, since we are already at the limit of what can be measured, no such thing can exist. Hence the order in which we measure the property is random.]
————–
If I were Wittgenstein I might have said:
Consider a situation in which someone asks, “How much light could you see?” Perhaps a detective is asking a hostage about where he was held. But then the answer is, “I didn’t look.” —— And this would make no sense.
hmmmm…. I did really mean to get back to gambling.
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07.29.08
Posted in Relativity, independence friendly logic, logic, measurement, philosophy, physics, science at 8:40 pm by nogre
Ever have the experience of sitting in traffic and believe that you are moving in reverse, only to realize a second later that you were fooled by the vehicle next to you moving forward? You were sitting still, but because you saw something moving away, you mistakenly thought you started to move in the opposite direction.
Two different senses may be at work here: your sight and your balance. Lets assume that your balance did not play any role in this little experiment (you would have been moving too slowly to feel a jolt). Your sight told you that you were moving in a certain direction (backwards) because of something you saw, say a bus pulling forward. Then you saw something other than the bus, say the ground, and you realized that your initial appraisal of the situation was incorrect.
At the point when you look away from the bus, you believe that you are moving backwards. Then when you see the ground, you believe that you are not moving backwards. You reconcile these two contradictory beliefs by deciding that it was not you who were moving backwards but the bus that was moving forwards.
What this illustrates is that objects require something other than themselves to be considered in motion. Without the ability to reference a ’stationary’ system (the ground), it is impossible to make a determination who is moving and who is staying still.
Now imagine this situation was taking place in a very gray place. The only things visible are yourself and the bus on a gray background. Then you notice that the bus is getting smaller. There is nothing for you to use as a reference (no stars, no ground, no nothing) to decide if it is you who is moving away from the bus or if it is the bus moving away from you, or both*. The only thing you have is the information that you and the bus are moving away from each other.
I refer to the statement that you and the bus are moving away from each other as information and not a belief because it is much more certain than what I called beliefs above, namely that you were in a certain kind of motion, which quickly turned out to be questionable.
The information that you and the bus are moving away from each other is not your everyday sort of information. It would be inaccurate to reduce this statement to a conjunction (you and the bus are moving), which is incorrect, or a disjunction (you or the bus is moving) because you are only moving with regard to the bus. By claiming that either you or the bus is moving, it makes it seem that the motion of one has nothing to do with the other. The motion of you and the bus need to be mutually dependent upon each other, and a mutual interdependence is not reducible.
If we return to the everyday, we can say that you have the information that you and the bus are moving away from each other and you and the bit of ground you are on are not moving away from each other. Since the bit of ground we initially selected was arbitrary (we could have chosen anything, like another bus) it is subject to the same issues as the bus; we merely take the ground to be stationary for most purposes, but this is a pragmatic concern. Hence all determinations of motion (or non-motion) are instances of informational interdependence.
The result that relativity is part of a larger class of mutually interdependent structures is non-trivial. Minimally this formalism will allow us to specify exactly when the use of relativity is warranted, but more importantly it will allow us to identify and provide insight into other situations of informational interdependence. Cases of mutual interdependence are relatively rare as far instances of logic go (they can’t even be described in first order logic) and having such a well studied example gives us a head start on this phenomenon.
—————————————-
* or if the bus is shrinking, or you are growing, or all of the above, but lets assume no Alice in Wonderland scenarios.
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04.26.08
Posted in Relativity, fun, game theory, independence friendly logic, internet, logic, philosophy at 2:59 pm by nogre
I picked up Dependence Logic: A New Approach to Independence Friendly Logic by Jouko Väänänen. I figure I’ll write up a review when I am finished with the book, but there is one chief difference between Dependence Logic and Independence Friendly Logic that needs to be mentioned.
On pages 44-47 when describing the difference between Dependence Logic and Independence Friendly Logic Väänänen says,
The backslashed quantifier,
∃xn\{xi0,…,xim-1}φ,
introduced in ref. [20], with the intuitive meaning:
“there exists xn, depending only on xi0,…,xim-1, such that φ,”
…
The slashed quantifier,
∃xn/{xi0,…,xim-1}φ,
used in ref. [21] has the following intuitive meaning:
“there exists xn, independently of xi0,…,xim-1, such that φ,”
which we take to mean
“there exists xn, depending only on variables other than xi0,…,xim-1, such that φ,”
The backslashed quantifier notation is part of what Väänänen calls ‘Dependence Friendly Logic’, and is equivalent to the ‘Dependence Logic’ that the rest of the book expounds. This backslash notation makes the difference between Dependence (Friendly) Logic and Independence Friendly Logic clear by showing that the former logic takes the notion of dependence to be fundamental whereas the latter takes independence to be fundamental. Väänänen takes this to be an advantage because he says that Dependence Logic avoids making
one ha[ve] to decide whether “other variable” refers to other variables actually appearing in a formula ?, or to other variables in the domain…
However, this treatment misses an important philosophical difference between Independence Friendly Logic and Dependence Logic. Dependence Logic is fundamentally based upon Wilfrid Hodges work, ‘Compositional Semantics for a language of imperfect information’ in Logic Journal of the IGPL (5:4 1997) 539-563, in which Hodges lays out a compositional semantics for languages such as Independence Friendly Logic using sets of assignments instead of individual assignments to determine satisfaction (T or F). Väänänen infers that Independence Friendly logic is just a bit unruly when it comes to specifying variables because he is working within a system that assumes sets of assignments are a useful and unproblematic way to determine satisfaction.
However the unseen problem of using sets of assignments is that something is added by assuming the domain is a set. For example, let’s take try to define a location and take the set of all the points in the universe. However, we immediately run into relativity: All locations are defined relative to each other and the people trying to figure out where things are, i.e. There is no predetermined set of all the points in the universe. The issue is that the domain of potential assignments, the objects in the universe, may be dependent upon the person or people using them (the players of the semantic game in this case). If the domain is dependent upon the players, the set cannot be constructed until after the players have begun the game. Therefore, if we postulate that the domain is a set at the outset then the players know something about the game that they are playing, namely that it does not depend upon them because it was predetermined.
Following this line of thought it seems possible to constructed a game in which the domain {Abelard, Eloise} is such that Abelard and Eloise are the actual people playing the game and the formula is ‘Someone x lost the game by instantiating this formula’ such that whoever instantiated that formula would win the game according to the rules. But then the formula would not be satisfied, so that player would have lost, but then it would be satisfied, a paradox. It is easy enough to declare that the domain must be independent of the players, but again this signals something about the game being played to the players before the formula to be is revealed.
Lastly there is something to be said about using logic to represent natural language here too: if you consider the set of all possible responses to some question, you are not ever considering all possible responses, but all the possible responses you can think of at that time. Therefore if we are using game semantics and imperfect information to represent natural language, then it is a mistake to predetermine the domain of all possible responses separate from the people involved. Again, the domain being linked to the people involved is at odds with the domain being a predetermined set.
Long story short, there is a very good reason for not always using sets of assignments to determine satisfaction. Depending on the situation, a set may offer non-trivial information about a game or misconstrue the game being played. Independence Friendly logic makes no assumptions about the type of game being played and is therefore of greater scope than logics that are based upon Hodges work. Of course one is free to use sets of assignments to determine satisfaction and derive set-theoretic results, but the compositionality gained comes at the price of limiting the types of games that can be played.
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02.06.08
Posted in Relativity, logic, philosophy at 7:49 pm by nogre
Apropos my earlier rant on people who think that paradoxes are meaningless, I figured I ought to take a stab at giving some meaning to paradox. To this end I reformulated a paradox in my terms. I suppose I should called it the Mirror Paradox, though ‘Looking-Glass Paradox’ seems more lyrical and has an historical nod. My apologies to whoever actually came up with this first, though I am sure I haven’t heard it before…
In my room I have a full length mirror. If I look at the man in the mirror and point to him saying, “There I am!” then where am I? If I am the man on the other side of the mirror, then I am not sitting on this side of the mirror. However, the man on the other side of the mirror has just pointed at me and said that he is not on his side of the mirror, but on mine. So I am not on my side, nor is he on his side. But then neither of us are on our side or on our mirror self side.
Now with semantic paradoxes and the like, we don’t have an agreed upon framework for analyzing what is going on in a paradox. Many times it is a paradox that signals that some such theory is unsatisfactory. However, this paradox deals with locations of people, namely me and mirror me, and we do have a general consensus on judging objects’ locations: in physics we determine some object’s location with reference to some previously agreed upon location.
Let us ignore for the moment that mirrors do not actually open up into other dimensions that you could enter if only your reflection didn’t get in the way. All that is important is that we have an exact double of yourself that at the instant you declare that you are where he or she is, that person does the exact same thing.
Declaring your location relative to your reflection is no different than declaring your location relative to anything else. Your reflection simultaneously declaring its location relative you is likewise unproblematic on its own. However, since the two non-identical perspectives are associated with only one person, we have a disconnect between perspectives and the person who holds the perspectives.
This problem of perspective is most telling. In Russell’s Paradox, there is no problem, no obvious contradiction that is, until the question “Is this set self-membered?” has been asked and answered twice. The first time through is arbitrary, let us assume no: Russell’s set is not part of itself. Now we ask, “If it is not self-membered, then is this set not self-membered?” Now we answer yes and have arrived at a contradiction. There is no problem yet, we merely have to revise our assumption: let us assume that Russell’s set is included in itself. Of course, then we ask, “If it is self-memberd, then is this set not self-membered?” and our answer is no: contradiction. At this point the paradox exists, but not before. It required us to look at the one set from two perspectives, one in which it is self-membered and one in which it is not.
The comparison of assumptions and perspectives that is drawn here is a good one. Our perspective, in a different sense, is our background assumptions. When we have contradictory perspectives on a subject we have incompatible background assumptions. The Mirror Paradox pulls our background assumption of location out of the background. We all assume that one perspective is associated with one location, but when you declare that you are someplace else and your reflection does the same, then you end up with two perspectives.
We can’t tell before hand whether we can have more than one perspective or a set that is defined by non-self-membership. Therefore, since the problem occurs with the selection of assumptions or perspective, the meaning of paradoxes, semantic or otherwise, is that your fundamental background assumptions are problematic. Sure, each paradox will only pertain to that particular system that it exists in, but for that system it will signal the most important and deep underlying problems.
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A side note: I thought of this while in bed last night and didn’t look at a mirror until this afternoon, even though I do have that full length mirror. Then I actually did point and say “There I am!” It was a bit of a strange experience because for some fraction of time I felt like me and my mirror self were in some sort of vortex with the rest of the world frozen outside. Almost needless to say I was a bit surprised if not shocked- I wasn’t expecting a reaction. When philosophy grabs you, even for an instant, it is spooky. I suggest you try this and see if you have the same reaction if only because I don’t think there are other paradoxes to actually participate in, save becoming a very methodological barber. How often do you get to participate in an experiment that isn’t prefaced by ‘thought’? Between the small mirror in my bathroom and my full length mirror, the full length elicited a better reaction, so use a full length one if you can.
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