Was Helmholtz a Bayesian? A review


 Justina Kennedy
 1 years ago
 Views:
Transcription
1 Perception, 2008, volume 37, pages 000 ^ 000 doi: /p5973 Was Helmholtz a Bayesian? A review Gerald Westheimer Division of Neurobiology, 144 Life Sciences Addition, University of California at Berkeley, Berkeley, CA , USA; Received 22 December 2007; in revised form 15 January 2008 Abstract. Modern developments in machine vision and object recognition have generated renewed interest in the proposal for drawing inferences put forward by the Rev. Thomas Bayes (1701 ^ 1759). In this connection the epistemological studies by Hermann Helmholtz (1821 ^ 1894) are often cited as laying the foundation of the currently popular move to regard perception as Bayesian inference. Helmholtz in his mature writings tried to reconcile the German idealist notions of realityashypothesis with scientists' quests for the laws of nature, and espoused the view that we ``attain knowledge of the lawful order in the realm of the real, but only in so far as it is represented in the tokens within the system of sensory impressions''. His propositions of inferring objects from internal sensory signals by what he called `unconscious inferences' have made Helmholtz be regarded as a protobayesian. But juxtaposing Bayes's original writings, the modern formulation of Bayesian inference, and Helmholtz's views of perception reveals only a tenuous relationship. ``Ich bin Bayesianer'' (Stegmu«ller 1973, page 117) 1 Introduction A large body of scholarship has grown around what is now called Bayesian statistics. The approach is widely used in, for example, machine vision and object recognition, and researchers in perception are being strongly urged to embrace it (Kersten et al 2004; Knill and Richards 1996). To enhance the argument, the name of Hermann Helmholtz (1821 ^ 1894), the founder and still most prominent contributor to visual science, is often invoked as having been a pioneer in thinking along the lines of modern Bayesian inference. To examine this claim it is necessary to study what both Bayes (1701 ^ 1759) and Helmholtz (figures 1 and 2) had actually written, as contrasted with the views now being imputed to them. Bayes's ideas, long in eclipse, had not yet resurfaced during Helmholtz's lifetime, nor had they assumed their modern form until well into the 20th century (Dale 1999). Hence the question can be raised whether Helmholtz, in a prescient way, had come to hold views that nowadays would be called Bayesian, even though, as we shall see, there is no consensus that Bayes actually held them himself in that form. 2 Modern formulation of Bayesian inference At present, one would call a Bayesian someone who subscribes to the standard formulation of Bayesian inference, as laid out in countless textbooks and articles. It runs something like the following (see, for example, MacKay 2003): Given an observed event E, and a set of hypotheses H 1 ; H 2 ; :::, then P(H 1 je ), the probability of the hypothesis H 1 given the event, is proportional to the product of two terms. The first is P(EjH 1, the probability of the event given that hypothesis H 1 ; it is called the likelihood. The second is P(H 1 ), the estimate that this hypothesis is in fact applicable, ie the prior probability. To reach an actual probability, a value between 0 and 1, a normalization factor is introduced to cover the full range of all hypotheses H 1, H 2, :::. Hence the usual formulation of the Bayes theorem nowadays is X P H 1 je ˆP E jh 1 6P H 1 P E jh i 6P H i Š. (1) i
2 2 G Westheimer Figure 1. The Rev. Thomas Bayes, F.R.S. (1701 ^ 1759) Figure 2. Exzellenz Hermann v. Helmholtz (1821 ^ 1894). Sometimes, a more particular formulation is used for cases in which there is a set of parameters w within the hypothesis H. For example, if H is the hypothesis that a Poisson distribution is involved, and w is the variance, the equation takes the following form X P w je, H ˆP E jw, H 6P w jh P E jw, H 6P wjh Š. The term P(E jw, H ) is unproblematic and involves the calculation of an inverse probability, viz the probability of the event E, given the hypothesis. Most of the discussion has centered on the origin and ultimate source of the `prior'. When applied to object recognition, the task is to decide whether a specific component in a captured picture is in fact the image of a real object. In this connection, the event is the presence in the image of a structure or a pattern that can be fully characterized, for example by the number and value of pixels. The hypothesis would be the existence of a specific object with a predefined structure in the real world. Enough would have to be known about the actual image and each of the hypothesized objects to calculate the probability that in the mode of transmission each of these objects would generate such an image. Also required are prior estimates of the probability of the objects being in fact out there in the real world. With this information it is possible to use the above equations to calculate the posterior probability that what has been received is in fact the image of a specific object. 3 What Bayes actually wrote It is of interest to compare the above formulation with what the Rev. Thomas Bayes had actually said. (1) In the posthumously published essay in Philosophical Transactions Bayes wrote that he was investigating how to solve the problem (Barnard 1958; Bayes 1763) (1) Bayes was not only accomplished as a mathematician but also had a deep understanding of the place of mathematics visa vis the natural world: ``It is not the business of a Mathematician to show that a strait line or circle can be drawn, but he tells you what he means by these; and if you understand him, you may proceed further with him; and it would not be the purpose to object that there is no such thing in nature as a true strait line or perfect circle, for this is not his concern: he is not inquiring how things are in matter of fact, but supposing things to be in a certain way, what are the consequences to be deduced from them; and all that is to be demanded from him is, that his suppositions be intelligible, and his inferences just from the suppositions he makes.'' (cited in Barnard 1958, page 294)
3 Was Helmholtz a Bayesian? 3 ``Given the number of times in which an unknown event has happened and failed: required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named.'' (page...) What follows has been described by one of its foremost interpreters as ``one of the most difficult works to read in the history of statistics'' (Stigler 1982). Bayes deals with the location along a distance AB of the point in which a thrown ball comes to rest, given that ``there shall be the same probability that it rests upon any one equal part of the plane as another, and that it must necessarily rest somewhere upon it''. First, a ball W is thrown once, landing in o (figure 3), but the position of o is not determined. What has been determined, though, is that a second similar ball, when thrown n times, landed p times to the right of o and q ˆ n p times to the left. Bayes is interested in the probability of o lying between any arbitrary points F and B along AB. In each ball throw ``there shall be the same probability that it rests upon any one equal part of the plane as another'' and ``the happening or failing of an event in different trials are so many independent events''. f b f b B o o A x Figure 3. Bayes considers the situation of a ball being thrown to land anywhere with equal probability along the distance AB, calling the event a success if it comes to rest to the right of a fixed point o whose position is unknown. The law governing the event is the binomial distribution with parameter x if one stipulates the event to have taken place on p occasions out of p q ˆ n trials. Bayes then goes on to calculate the chance of p successes in n trials for the interval between two points f and b within AB. It is the ratio of the area under the likelihood distribution of binomials between parameters f and b to that under the whole distribution between A and B. Bayes calls the situation in which a ball lands to the right of o as ``the happening of the event M'' and argues that its occurrence p times in p q ˆ n trials is governed by the binomial distribution (n!=p!q!)6x p 6(1 x) q where x is the ratio of the distances Ao and AB. In modern parlance, the ball coming to rest to the right of o is the event and the binomial distribution in the hypothesis. Bayes then presciently used what is now known as the inverse approach: For any other point along AB, say f, the chance that a ball tossed n times would on p occasions land to the right of f is also governed by the binomial, now (n!=p!q!)6f p 6 1 f ) q, where f is the distance Af. Hence the desired answer, viz the chance that o lies between f and B, is given by the sum of the binomials with all the parameters between f and B. In order to arrive at a probability, ie a value between zero and one, this would have to be divided by the sum for all points between A and B. According to Bayes, this is also the probability that a single throw from the same ensemble lands in that interval. Bayes approach is to identify the event E and the governing hypothesis w, H (that the chance of a tossed ball landing to the right of any dividing point of a line of unit length is given by the binomial distribution with the parameter w, the ratio of the P dividing point distance to the length of the whole line) and then to compute P E jw, H, the chance of the event occurring for the stipulated parameter range of the hypothesis. In other words, Bayes's operations take place in the realm of likelihood.
4 4 G Westheimer Just to illustrate what Bayes was after: If 5 out of 25 ball tosses gave a positive answer to the question whether the event has occurred, what is the probability that o lies in the middle 10% of the line AB, ie between f ˆ 0:45 and b ˆ 0:55? If B (x; n, p) ˆ n!= p!(n p)!š6x p 6(1 x) q, then the required probability is x ˆ 0:55 x ˆ 1:00 P P B x; 25, 5 B x; 25, 5 x ˆ 0:45 x ˆ 0 The argument proceeds along the following lines: For x ˆ 0:45, what is the probability that a single throw lands to the right of it? Here n ˆ p ˆ 1 and (0:45; 1, 1) ˆ 0:45. For one throw out of two it is B 0:45; 2, 1 ˆ0:495, because either both throws go into one or the other division of the whole distance, with probabilities of (0.45) 2 and (0.55) 2 respectively, or one throw goes into each, with probability 0:4560:55 ˆ 0:2475, which can occur in two ways. Finally, for exactly 5 throws out of 25, the binomial value B(0:45; 25, 5) is equal to ; P B x; 25, 5 for the range 0.45 to 0.55 computes to and for the full distance 1 4 x 4 0 to Hence, in this particular application of Bayes's problem, namely `If 5 out of 25 tosses had a positive outcome, what is the probability of 0:55 4 o 4 0:45?' the answer is =3.846 ˆ , or 1 in 154. On the other hand, for 0:3 4 x 4 0:2 itis0.453or1in2.2. But it is the prior that is the crucial component of all the Bayesian discussions. In Bayes's words, in arriving at the value of his expectation he wanted to consider whether there was anything that may give ``reason to think that, in a certain number of trials, [the event] should rather happen any one possible number of times than another''. Even here, unfortunately, Bayes's writing has led to debates. Having called the outcome of the ballthrow the ``event M'' in prop. 9 where he lays out the binomial proposition, he goes on to say: ``In what follows therefore I shall take for granted that the rule given concerning the event M in prop. 9 is also the rule to be used in relation to any event concerning the probability of which nothing at all is known antecedently to any trials made or observed concerning it.'' (cited in Barnard 1958, page 306). Such a phrasing might give the impression that Bayes rejected the idea of a prior, but this is not the case, for he states explicitly concerning both the fiduciary and test throws ``that if either... be thrown there shall be the same probability that it rests upon any one equal part of the plane as another'', ie the prior is flat. Stigler, who subjected the situation to the most thorough critical analysis, concluded that for Bayes the prior is associated with the locations of the subsequent throws and not those leading to the formulation of the hypothesis H, as is now embodied in the canon of Bayesian inference. As it happens, in the situation analyzed by Bayes, both have a flat prior and the distinction is not material. However, the substitution of P(E ) for P(H ) in equation (1) would reformulate Bayes's theorem into P H je ˆP E jh 6P E X P E jh 6P E Š and deprive it of its main attraction, namely the ability to assign a different probability to one hypothesis or model, or one range of parameters within a hypothesis, than to others. The starting point of modern Bayesian analyses is a particular set of observations and the assignment of higher probabilities to some than to others surely must relate to hypotheses and not observations.
5 Was Helmholtz a Bayesian? 5 Thus is seems that, if being a Bayesian means being a strict follower of Bayes, it does not necessarily imply unconditional acceptance of the current interpretation of all the terms of equation (1), even if there were no other challenges to their application. Bayes was ignored for at least a century and a half. When his work was resurrected, it was in the context of arguments between warring factions in statistics and probability theory who tried to come to terms with the extremely deep problem of defining probability. The situation is one in which Einstein's famous dictum (made in connection with geometry) might be rephrased: Insofar as probability definitions are mathematical, they do not refer to reality; insofar as they refer to reality they are not mathematical. In mathematics one looks for proofs of convergence of infinite series. But in situations where probability is invoked, even in as long a sequence of dice throws (or balls drawn from an urn with replacement) as might be contemplated, the outcome is ultimately uncertain and hence unsatisfactory in the strictest logicodeductive environment of mathematics. Again and again, in probability theory and statistics, recourse has to be sought in how the individual regards chance and would deal with it. Bayes writes about expectation, gain and loss, and it is not by accident that the one surviving original document of his was found in the archives of an insurance company. 4 Helmholtz's epistemology No single person, before or since, contributed more to the knowledge of the human sensory apparatus than Hermann Helmholtz, and throughout his career he kept concerning himself with questions of the origin of our visual experiences. He first broached the subject in an 1854 lecture, as a 34yearold beginning professor of physiology in Ko«nigsberg, and returned to it in a variety of settings till almost the last essay he wrote during the year of his death in The introduction to part III of the first edition of the Handbuch der physiologischen Optik, which forms the basis of the English version, contains a sentence which best encapsulates the views most widely attributed to him. ``The general rule according to which visual representations determine themselves... is that we always find present in the visual field such objects as would have to exist in order for them to produce the same impression on the neural apparatus under the usual normal conditions of the use of our eyes.'' (Helmholtz 1867, page 428/1911, page 4) (2) In gauging the evolution of Helmholtz's views, however, it is important to realize that this passage was omitted in the second edition (published in 1895, but not available in an English translation). In its place Helmholtz wrote an extensive revision of the section which is more in line with the pivotal formulations to which we now turn. The roots of Helmholtz's epistemological writings are twofold. First, he was one of the ablest and most consequential practitioners of midnineteenth century natural scienceöand by far the most successful of those in sensory physiologyöwho erected a securely structured body of knowledge based on relatively uncomplicated empirical observations and deductive rules of manageable mathematical complexity. Second, he was firmly grounded in the academic and cultural (and later even the industrial) elite of that other solidly and successfully constructed element of nineteenthcentury European history, the Prussian state. The two streams converged in 1878, when as Rector of Berlin University he took the occasion of the solemn Founder'sDay address to present the credo of his epistemological system. The speech represented a synthesis of (2) The wording of the passage differs somewhat from the one in Southall's English translation which, as many of the English versions of Helmholtz's writing, does not always capture the subtleties of Helmholtz's phrasing. All quotes in this paper have been rendered into English by the author.
6 6 G Westheimer the aims of a working scientist in the area of sensory perception and those of a loyal member of an intellectual community in which Kant and Goethe were revered. (3) Its very title affords an illustration of the difficulties encountered in viewing Helmholtz entirely via the English version of his writings. Helmholtz toyed with various other titles (Koenigsberger 1903), such as ``Prinzipien der Wahrnehmung'' (Principles of perception), ``Was ist wirklich?'' (What is real?), and a favorite citation from Goethe's Faust `Àlles Verga«ngliche ist nur ein Gleichniss'' (All that is transitory is only a metaphor). In the end he chose ``Tatsachen in der Wahrnehmung'' which surely should be rendered ``Facts in perception'' and not, as a prominent version would have it, ``The facts of perception''. He quickly cut to the chase: ``What is truth in our percepts and thoughts? In what way do our ideas correspond to reality? Philosophyöit tries to sift what in our knowledge and ideas is due to the influence of the material world in order to establish what belongs to the purely innate activity of the mind. Scienceöit, to the contrary, tries to sift what is definition, notation, manner of representation, hypothesis, in order to lay out, in a pure form, what belongs to the world of reality whose laws it seeks.'' (Helmholtz 1878/1903, page 218) In order to mediate between them, Helmholtz first of all recognizes and accepts the validity of the program of the two camps over which he as Rector was presiding. And no one was better suited for the mediating task than this iconic presence in natural science who at the same time could boast as family acquaintance the founding philosopher of German idealism, and predecessor in the rectorate a halfcentury earlier, Johann Gottlieb Fichte. Helmholtz was driven to his cogitations when, after sequentially analyzing the optical, anatomical, physiological, and psychophysical stages of vision, he confronted the nature and origin of perception. He realized that the physicalist/materialist approach that guided him through the physics and biology thus far had come to the end of its rope, here in the borderland between the material and mental worlds. Seeking guidance in Kant's writing, he could not refute that philosopher's proposition that a deep chasm divides the concept of an object's representation in the mind from that object's hypothesized existence in the outside world. Helmholtz was a medical graduate, a sometime Professor of Physiology at Heidelberg, now head of one of the great departments of Physics, and he owed it to his constituents in these professions and to his own scientific achievements to hold fast to the concept of a reality in which the laws articulated by natural science held sway. At the same time he was thoughtful and thoroughgoing and could not, therefore, ignore Kant's teaching. The reconciliation is elegant: ``The distinction between thought and reality is possible only when we know how to distinguish between what the `I' can change and what the `I' cannot change.... What we then attain is knowledge of the lawful order in the realm of the real, but only in so far as it is represented in the tokens within the system of sensory impressions.'' (Helmholtz 1878/ 1903, page 242) The phrasing becomes more impressive when it is realized that `I' and `noti' (`Ich' and `NichtIch') are essential formulations in Fichte's writing, who counterposed the incarnations of mind, will, faith, morals, to those of the material and of nature (Schmidt 1969) (3) Helmholtz's standing in the German Imperial establishment is attested to in a memo from the Minister of Education to Chancellor Bismarck relating to the appointment to the presidency of the new Imperial Bureau of Standards. Helmholtz should maintain his association with the University because it ``would retain a man who has been viewed for many years as its scientific head, who contributed more than anyone else to smooth out... the contrast between the natural sciences and the humanities, and who, in the arena of politics, fostered the moderately conservative tendency in which Berlin University is well in advance among German Universities'' (Koenigsberger 1903, page 353). In 1883 he was raised to the hereditary nobility and in 1891 he became Privy Councilor to be addressed `His Excellency'.
7 Was Helmholtz a Bayesian? 7 and who, as an idealist philosopher, even went so far as to posit that the `I' generates the `noti'. All the ingredients of Helmholtz's argument are contained in the paragraph: there is a realm of the real with its lawful order; what we know about it is represented by tokens within the system of sensory impressions; and finally, our knowledge hinges on active explorationöreality is what remains invariant when the expected changes due to willed movements are factored out from the sensory impressions. 5 Helmholtz on inference The single most enveloping aspect of Helmholtz as a scientist and epistemologist was a belief in empiricism which he espoused throughout his career. Specifically, he continued to assert that our knowledge of the real world is derived by using our motor system as exploring organ to deduce invariances by trial and error. Generating a movement is the activity of the `I' which keeps track of the instructions. (A modern term is efference copy, meaning the record that is maintained of the outgoing or efferent signals from the central nervous system to the muscles.) The associated change in sensory signals is registered, and inferences can be drawn from a `before' and `after' comparison. Through knowledge of the actuated movement it can be determined what in the changes of the sensory impressions can be ascribed to the movements; what remains, by inference, is of the real world. Such inferences are the same as the deductions in the realm of ordinary logic, only here we are not aware of them and hence the appellation `unconscious inferences'. Helmholtz's clearest and most specific illustration of how he imagines this process to operate is in the arena of retinal local signs. He agrees with Lotze that each location of the retinal periphery has its own spatial value. In the fully developed organism a nexus has been established between this spatial value and the eye rotation necessary to foveate a target imaged on that peripheral retinal location. When a willed eye rotation has been executed, and both its extent and the shift in retinal location have been registered internally, the spatial location and extent of the object can be inferred. Helmholtz here has satisfied the imperatives both of the natural scientist of his day, to whom a real world was a given, and of Kantian thinkers, for whom the `Ding und sich' is in principle unreachable: the real world was a hypothesis. As a practicing scientist he is, of course, obliged to argue that there is nothing wrong with hypotheses, and in fact the whole enterprise of science restsöquite firmlyöon this premise: ``In its essence, each properly constituted hypothesis proposes a more general law of phenomena than we had obtained by immediate observations up to then, and is an attempt to rise to an ever more general and comprehensive set of laws. Any new facts asserted by such a hypothesis must be tested and affirmed by observations and experiments.'' (Helmholtz 1878/1903, page 242) He will not be pushed into a corner by absolutists and pure materialists: ``Any reduction of phenomena to underlying causes and forces asserts that we have found something permanent and final. Such an unconditional statement is, however, never justified; the incompleteness of our knowledge does not allow this, nor does the nature of conclusions from inferences on which, right from the beginning, our perceptions of the real are based.'' (page 243) Nor is he unaware of the problem of causality that had exercised Hume's mind: ``Every inductive inference is based on the trust that previously observed lawful behavior will be found valid in all cases that have yet to be observed. This is the trust in the lawfulness of all phenomena. But that lawfulness is the condition of comprehensibility.... The law of causality expresses a trust in the complete comprehensibility (vollkommende Begreifbarkeit) of the world. Comprehending, in the sense in which I have described it, is the method used by our thinking to subdue the world, to order facts, to predict the future.
8 8 G Westheimer It has the right and the duty to extend its method to occurrences, and it has indeed harvested great results in this way. For the utilization of the law of causality, however, we have no guarantee other than its success.'' (page 243) 6 Can Helmholtz's proposition be called Bayesian? Helmholtz routinely made the distinction between sensation (Empfindungen) and perception (Wahrnehmung). It is in connection with perception that he sponsored the proposition that it is based on sensory tokens and motor outflow signals, both internal to the organism, from which inferences are drawn about objects. These inferences are constantly updated and affirmed by further probing. Helmholtz's pronouncements were made in a general philosophical context; he was actually much more measured when offering specific instances. He argued all along that many of our visual abilitiesöspatial localization for exampleöare not inborn but acquired, and it was therefore incumbent on him to explain how they arose. When he employed the word `object' it was to refer to an elemental stimulus such as a specific target location or object point rather than to a visual feature such as a person or a tree. Still, in being explicitly on record that the real world and its laws are hypotheses inferred from internal sensory signals, Helmholtz's view of the process of perception does indeed have some similarity with the process utilized in machine vision in the attempts to identify objects from computer images. Does this make him a Bayesian? The Bayesian approach asks first of all: What is the probability of the event, given the hypothesis, P(E jh. Then, to arrive at the posterior probability of the hypothesis H, given the event E, ie P(H je, P(E jh is multiplied by a prior probability of the hypothesis, P(H ). If the object is the hypothesis, and the sensory tokens and the motor outflow signals are the event, then the act of perception is the inference of H given E, and the endpoint of the perceptual process is indeed equivalent to the endpoint of a Bayesian process. Thus far there is no dissonance. But what about the intervening steps? Does Helmholtz propose a stage in which the hypotheses (possible objects) are each tested against the event, ie the likelihoods are estimated so that each of an array of objects could have given rise to the particular set of sensory and motor signals experienced by the observer? In other words, does he believe that an internal record is available of each of the possible objects (ie hypotheses), for purposes of computation of its P(E jh )? And further: Can some analog be found in Helmholtz's proposition for priors, ie the availability of P(H ) to be associated with each P(E jh )? Although there are many signs of its evolution over his lifetime, Helmholtz's epistemological writings are remarkably consistent, and in none of them can one find substantiation that he was willing to frame the problem of perception in such a form. Presentday Bayesian statistics is, of course, not at all the same as it was at the time is was formulated in the midde of the 18th century. Helmholtz's work dates back before the modern era of Bayesian statistics and, as we have seen, Bayes himself gave much less emphasis to `priors' than his current disciples. Could Helmholtz perhaps be said to have been a Bayesian in the more original form? A reading of the original, cited above, makes the connection seem rather tenuous between Bayes's calculation of the probability for the location of a ball thrown on a table and Helmholtz's proposition for inferring the location of an object in the real world from sensory signals. Yet one can conjecture that, had he been shown the modern Bayesian equation and confronted with the challenge to update his writing, Helmholtz would not have abjured the proposition that hypotheses about real objects, based on the internal representations in our sensory signals, are developed by some mechanism involving inverse probabilities and priors. He agonized about these issues 150 years ago, being torn between
9 Was Helmholtz a Bayesian? 9 the opposing imperatives of the natural scientist trying for laws of reality, and the philosopher in quest for the ultimate source of our knowledge, He understood perhaps better than any of his contemporaries how this division of labor plays out: ``The natural sciences still rest on the same firm foundations that they had in Kant's days and whose productive utilization was best exemplified by Newton.... Kant's philosophy did not have the intention of augmenting the amount of our knowledge by means of pure thought because its prime tenet is that knowledge of reality must emerge from experience; its sole purpose was to examine the source of our knowledge and its justification, something that will always remain philosophy's task and that no generation can shirk with impunity.'' (Helmholtz 1855/1903, page 88) As Helmholtz understood, the ``firm foundations of natural sciences'', in their exemplary ``productive utilization... by Newton'', they involved observations and the testing of hypotheses. This holds now as it did then and, as everywhere else, in the arena of perception the enterprise rests on observables. Object recognition by machine vision tries to relate images, which are observable, with realworld objects which, though they are formulated in terms of hypotheses, are nevertheless in the realm of the tangible. Neurophysiological exploration utilizes known objects and tangible records of neural activity; the hypotheses about their relationship, if intended to have `productive utilization', have to be testable. Bayesian inference without a doubt has a defined place in these endeavors, which are often covered by the phrase `empirical Bayesian'. But the motto ``Perception as Bayesian Inference'' might be read as implying that human visual perception is a Bayesian act of inferring a hypothesized reality based on internal sensory impressions and on memory traces of a different kind of previous experiences and their frequency. Of course, as Rector of Berlin University, Helmholtz would not, and as a deeply consequent thinker could not, refute Kant's imperative of a transcendental reality. But he was alsoöand foremostöa natural scientist. In contrast with philosophical rumination about the immediacy of an Anschauung and the inferred nature of reality, research in human perception cannot but follow the procedures in the rest of natural science which lead from observables to hypothesized states, with the laws linking the two devised so as to be testable. If ``Perception as Bayesian Inference'' is taken to mean that the percept is the event and the object is the hypothesis, such a program to be viable needs the percept to be moved into the realm of the observable. The challenge to achieve this has been with the discipline at least since the advent of the Gestalt movement. If he lived now, Helmholtz would have congratulated the machinevision community on what they are achieving, but as a student of the human visual apparatusöwilling as he might be to acknowledge the primacy of mental perceptual processes and to posit reality as hypothesisöwould have concentrated on the more traditional research methodology, wherein propositions are tested by presenting targets in the real world and inferences made about the internal state of the organism. References Barnard G A, 1958 ``Thomas Bayes's essay towards solving a problem in the doctrine of chances'' Biometrika ^ 315 Bayes T, 1763 `Àn essay towards solving a problem in the doctrine of chances'' Philosophical Transaction of the Royal Society of London ^ 418 Dale A I, 1999 A History of Inverse Probability from Thomas Bayes to Karl Pearson (New York: Springer) Helmholtz H von, 1855/1903 ``Ueber das Sehen des Menschen'', in Vortra«ge und Reden (Braunschweig: Vieweg) pp 85 ^ 117 Helmholtz H von, 1867 Handbuch der physiologischen Optik (Leipzig: Voss) Helmholtz H von, 1878/1903 ``Die Tathsachen in der Wahrnehmung'', in Vortra«ge und Reden (Braunschweig: Vieweg) pp 213 ^ 247
Honours programme in Philosophy
Honours programme in Philosophy Honours Programme in Philosophy The Honours Programme in Philosophy offers students a broad and indepth introduction to the main areas of Western philosophy and the philosophy
More informationArgument Mapping 2: Claims and Reasons
#2 Claims and Reasons 1 Argument Mapping 2: Claims and Reasons We ll start with the very basics here, so be patient. It becomes far more challenging when we apply these basic rules to real arguments, as
More informationON EXTERNAL OBJECTS By Immanuel Kant From Critique of Pure Reason (1781)
ON EXTERNAL OBJECTS By Immanuel Kant From Critique of Pure Reason (1781) General Observations on The Transcendental Aesthetic To avoid all misapprehension, it is necessary to explain, as clearly as possible,
More information1/9. Locke 1: Critique of Innate Ideas
1/9 Locke 1: Critique of Innate Ideas This week we are going to begin looking at a new area by turning our attention to the work of John Locke, who is probably the most famous English philosopher of all
More informationTHE KNOWLEDGE ARGUMENT
Michael Lacewing Descartes arguments for distinguishing mind and body THE KNOWLEDGE ARGUMENT In Meditation II, having argued that he knows he thinks, Descartes then asks what kind of thing he is. Discussions
More informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationREASONS FOR HOLDING THIS VIEW
Michael Lacewing Substance dualism A substance is traditionally understood as an entity, a thing, that does not depend on another entity in order to exist. Substance dualism holds that there are two fundamentally
More informationDescartes. Philosophy and Good Sense
Perspectives in Philosophy Rene Descartes Descartes Philosophy is the search for certainty the search to know, for yourself, what is really true and really false to know which beliefs are reliable. However,
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationTime and Causation in Gödel s Universe.
Time and Causation in Gödel s Universe. John L. Bell In 1949 the great logician Kurt Gödel constructed the first mathematical models of the universe in which travel into the past is, in theory at least,
More informationPhil 420: Metaphysics Spring 2008. [Handout 4] Hilary Putnam: Why There Isn t A ReadyMade World
1 Putnam s Main Theses: 1. There is no readymade world. Phil 420: Metaphysics Spring 2008 [Handout 4] Hilary Putnam: Why There Isn t A ReadyMade World * [A readymade world]: The world itself has to
More informationAbstraction in Computer Science & Software Engineering: A Pedagogical Perspective
Orit Hazzan's Column Abstraction in Computer Science & Software Engineering: A Pedagogical Perspective This column is coauthored with Jeff Kramer, Department of Computing, Imperial College, London ABSTRACT
More informationLecture 17 Newton on Gravity
Lecture 17 Newton on Gravity Patrick Maher Philosophy 270 Spring 2010 Introduction Outline of Newton s Principia Definitions Axioms, or the Laws of Motion Book 1: The Motion of Bodies Book 2: The Motion
More information1/10. Descartes 2: The Cogito and the Mind
1/10 Descartes 2: The Cogito and the Mind Recap: last week we undertook to follow Descartes path of radical doubt in order to attempt to discover what, if anything, can be known for certain. This path
More informationHow does the problem of relativity relate to Thomas Kuhn s concept of paradigm?
How does the problem of relativity relate to Thomas Kuhn s concept of paradigm? Eli Bjørhusdal After having published The Structure of Scientific Revolutions in 1962, Kuhn was much criticised for the use
More informationReality in the Eyes of Descartes and Berkeley. By: Nada Shokry 5/21/2013 AUC  Philosophy
Reality in the Eyes of Descartes and Berkeley By: Nada Shokry 5/21/2013 AUC  Philosophy Shokry, 2 One person's craziness is another person's reality. Tim Burton This quote best describes what one finds
More informationLecture 8 The Subjective Theory of Betting on Theories
Lecture 8 The Subjective Theory of Betting on Theories Patrick Maher Philosophy 517 Spring 2007 Introduction The subjective theory of probability holds that the laws of probability are laws that rational
More informationMethodological Issues for Interdisciplinary Research
J. T. M. Miller, Department of Philosophy, University of Durham 1 Methodological Issues for Interdisciplinary Research Much of the apparent difficulty of interdisciplinary research stems from the nature
More informationMathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades
Appendix A Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades To respond correctly to TIMSS test items, students need to be familiar with the mathematics
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationMind & Body Cartesian Dualism
Blutner/Philosophy of Mind/Mind & Body/Cartesian dualism 1 Mind & Body Cartesian Dualism The great philosophical distinction between mind and body can be traced to the Greeks René Descartes (15961650),
More informationThis is because the quality of extension is part of the essence of material objects.
UNIT 1: RATIONALISM HANDOUT 5: DESCARTES MEDITATIONS, MEDITATION FIVE 1: CONCEPTS AND ESSENCES In the Second Meditation Descartes found that what we know most clearly and distinctly about material objects
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationKant s Fundamental Principles of the Metaphysic of Morals
Kant s Fundamental Principles of the Metaphysic of Morals G. J. Mattey Winter, 2015/ Philosophy 1 The Division of Philosophical Labor Kant generally endorses the ancient Greek division of philosophy into
More informationUNIVERSALITY IS UBIQUITOUS
UNIVERSALITY IS UBIQUITOUS Martin Davis Professor Emeritus Courant Institute, NYU Visiting Scholar UC Berkeley Q 3 a 0 q 5 1 Turing machine operation: Replace symbol ( print ) Move left or right one square,
More informationDescartes Meditations Module 3 AQA. Meditation I Things which can be called into Doubt
Descartes Meditations Module 3 AQA Meditation I Things which can be called into Doubt Descartes rejects all his beliefs about the external world because they are doubtful and he wants to find a foundation
More informationDescartes : The Epistemological Argument for MindBody Distinctness. (Margaret Wilson)
Descartes : The Epistemological Argument for MindBody Distinctness Detailed Argument Introduction Despite Descartes mindbody dualism being the most cited aspect of Descartes philosophy in recent philosophical
More informationIs Justified True Belief Knowledge?
Is Justified True Belief Knowledge? EDMUND GETTIER Edmund Gettier is Professor Emeritus at the University of Massachusetts, Amherst. This short piece, published in 1963, seemed to many decisively to refute
More informationSubject area: Ethics. Injustice causes revolt. Discuss.
Subject area: Ethics Title: Injustice causes revolt. Discuss. 1 Injustice causes revolt. Discuss. When we explain phenomena we rely on the assertion of facts. The sun rises because the earth turns on its
More informationAn Innocent Investigation
An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@press.princeton.edu
COPYRIGHT NOTICE: Richard Raatzsch: The Apologetics of Evil is published by Princeton University Press and copyrighted, 2009, by Princeton University Press. All rights reserved. No part of this book may
More informationThe Null Hypothesis. Geoffrey R. Loftus University of Washington
The Null Hypothesis Geoffrey R. Loftus University of Washington Send correspondence to: Geoffrey R. Loftus Department of Psychology, Box 351525 University of Washington Seattle, WA 981951525 gloftus@u.washington.edu
More informationComparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Be able to explain the difference between the pvalue and a posterior
More informationLast time we had arrived at the following provisional interpretation of Aquinas second way:
Aquinas Third Way Last time we had arrived at the following provisional interpretation of Aquinas second way: 1. 2. 3. 4. At least one thing has an efficient cause. Every causal chain must either be circular,
More informationGeneral Philosophy. Dr Peter Millican, Hertford College. Lecture 3: Induction
General Philosophy Dr Peter Millican, Hertford College Lecture 3: Induction Hume s s Fork 2 Enquiry IV starts with a vital distinction between types of proposition: Relations of ideas can be known a priori
More informationFrege s theory of sense
Frege s theory of sense Jeff Speaks August 25, 2011 1. Three arguments that there must be more to meaning than reference... 1 1.1. Frege s puzzle about identity sentences 1.2. Understanding and knowledge
More informationThe Logical Way to Teach Introduction to Philosophy. Gabriel R. Camacho El Paso Community College, Transmountain Campus
1 The Logical Way to Teach Introduction to Philosophy Gabriel R. Camacho El Paso Community College, Transmountain Campus Correspondence concerning this article should be addressed to Gabriel R. Camacho,
More informationTHE SAL FORENSICS CONFERENCE 2009 THURSDAY, 8 OCTOBER 2009. Opening Remarks of Chief Justice Chan Sek Keong
THE SAL FORENSICS CONFERENCE 2009 THURSDAY, 8 OCTOBER 2009 Opening Remarks of Chief Justice Chan Sek Keong 1. Over the next two days, the Conference will involve experts in discussions on a broad range
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationIs a SingleBladed Knife Enough to Dissect Human Cognition? Commentary on Griffiths et al.
Cognitive Science 32 (2008) 155 161 Copyright C 2008 Cognitive Science Society, Inc. All rights reserved. ISSN: 03640213 print / 15516709 online DOI: 10.1080/03640210701802113 Is a SingleBladed Knife
More informationPsychology has been considered to have an autonomy from the other sciences (especially
THE AUTONOMY OF PSYCHOLOGY Tim Crane, University College London Psychology has been considered to have an autonomy from the other sciences (especially physical science) in at least two ways: in its subjectmatter
More informationBASIC STATISTICAL METHODS FOR GENOMIC DATA ANALYSIS
BASIC STATISTICAL METHODS FOR GENOMIC DATA ANALYSIS SEEMA JAGGI Indian Agricultural Statistics Research Institute Library Avenue, New Delhi110 012 seema@iasri.res.in Genomics A genome is an organism s
More informationSECTION 102 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationOn the Nature of Measurement in Quantum Mechanics. Abstract. Text
of Measurement in Quantum Mechanics DOUGLAS M. SNYDER LOS ANGELES, CALIFORNIA Abstract A number of issues related to measurement show that selfconsistency is lacking in quantum mechanics as this theory
More informationDELAWARE MATHEMATICS CONTENT STANDARDS GRADES 910. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))
Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 910) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS
More informationCRITICAL THINKING REASONS FOR BELIEF AND DOUBT (VAUGHN CH. 4)
CRITICAL THINKING REASONS FOR BELIEF AND DOUBT (VAUGHN CH. 4) LECTURE PROFESSOR JULIE YOO Claims Without Arguments When Claims Conflict Conflicting Claims Conflict With Your Background Information Experts
More informationThe Slate Is Not Empty: Descartes and Locke on Innate Ideas
The Slate Is Not Empty: Descartes and Locke on Innate Ideas René Descartes and John Locke, two of the principal philosophers who shaped modern philosophy, disagree on several topics; one of them concerns
More informationLEARNING OUTCOMES FOR THE PSYCHOLOGY MAJOR
LEARNING OUTCOMES FOR THE PSYCHOLOGY MAJOR Goal 1. Knowledge Base of Psychology Demonstrate familiarity with the major concepts, theoretical perspectives, empirical findings, and historical trends in psychology.
More informationA Few Basics of Probability
A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study
More informationSimplifying Bayesian Inference
Simplifying Bayesian Inference Stefan Krauß, Laura Martignon & Ulrich Hoffrage Max Planck Institute For Human Development Lentzeallee 94, 14195 BerlinDahlem Probability theory can be used to model inference
More informationBayesian probability theory
Bayesian probability theory Bruno A. Olshausen arch 1, 2004 Abstract Bayesian probability theory provides a mathematical framework for peforming inference, or reasoning, using probability. The foundations
More informationHistory of Western Philosophy: Descartes to Kant, Spring 2011
Marina Folescu / History of Western Philosophy / Sample Syllabus page 1 of 5 History of Western Philosophy: Descartes to Kant, Spring 2011 Instructor: Marina Folescu Email: folescu@usc.edu Homepage: http://wwwscf.usc.edu/
More informationIntroduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative
More informationWhat is Psychology? A set of questions about mental functioning trace back to philosophy Aristotle asked about memory, personality, emotions, etc.
What is? The science of behavior and the mind behavior  observable actions of a person or animal mind  thoughts, feelings, sensations, perceptions, memories, dreams, motives and other subjective experiences
More informationArnold Zellner. Booth School of Business. University of Chicago. 5807 South Woodlawn Avenue Chicago, IL 60637. arnold.zellner@chicagobooth.
H.G.B. Alexander Research Foundation Graduate School of Business University of Chicago Comments on Harold Jeffreys Theory of Probability Revisited, coauthored by C.P. Robert, N. Chopin and J. Rousseau.
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationLaplace's Demon. By finishing the work began by Sir Isaac Newton in mathematics, and further
Joshua Baker Philosophy 123 Professor Slinker Laplace's Demon By finishing the work began by Sir Isaac Newton in mathematics, and further applying it to physics and astronomy, French physicist and mathematician
More informationProbability Using Dice
Using Dice One Page Overview By Robert B. Brown, The Ohio State University Topics: Levels:, Statistics Grades 5 8 Problem: What are the probabilities of rolling various sums with two dice? How can you
More informationThe Special Theory of Relativity explained to children
The Special Theory of Relativity explained to children (from 7 to 107 years old) CharlesMichel Marle marle@math.jussieu.fr Université Pierre et Marie Curie, Paris, France Albert Einstein Century International
More informationIn Defense of Kantian Moral Theory Nader Shoaibi University of California, Berkeley
In Defense of Kantian Moral Theory University of California, Berkeley In this paper, I will argue that Kant provides us with a plausible account of morality. To show that, I will first offer a major criticism
More informationABA. History of ABA. Interventions 8/24/2011. Late 1800 s and Early 1900 s. Mentalistic Approachs
ABA Is an extension of Experimental Analysis of Behavior to applied settings Is not the same as modification Uses cognition in its approach Focuses on clinically or socially relevant s Is used in many
More informationRead Before You Cite!
Read Before You Cite! M. V. Simkin V. P. Roychowdhury Department of Electrical Engineering, University of California, Los Angeles, CA 900951594 We report a method for estimating what percentage of people
More informationWhat Is School Mathematics?
What Is School Mathematics? Lisbon, Portugal January 30, 2010 H. Wu *I am grateful to Alexandra AlvesRodrigues for her many contributions that helped shape this document. The German conductor Herbert
More informationIt has been contended that it would be possible for a socialist economy to solve
THE EQUATIONS OF MATHEMATICAL ECONOMICS AND THE PROBLEM OF ECONOMIC CALCULATION IN A SOCIALIST STATE LUDWIG VON MISES I It has been contended that it would be possible for a socialist economy to solve
More informationCompass Interdisciplinary Virtual Conference 1930 Oct 2009
Compass Interdisciplinary Virtual Conference 1930 Oct 2009 10 Things New Scholars should do to get published Duane Wegener Professor of Social Psychology, Purdue University Hello, I hope you re having
More informationWhat is Bayesian statistics and why everything else is wrong
What is Bayesian statistics and why everything else is wrong 1 Michael Lavine ISDS, Duke University, Durham, North Carolina Abstract We use a single example to explain (1), the Likelihood Principle, (2)
More informationThe result of the bayesian analysis is the probability distribution of every possible hypothesis H, given one real data set D. This prestatistical approach to our problem was the standard approach of Laplace
More informationQuine on truth by convention
Quine on truth by convention March 8, 2005 1 Linguistic explanations of necessity and the a priori.............. 1 2 Relative and absolute truth by definition.................... 2 3 Is logic true by convention?...........................
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationIntroduction. Hegel s Trinitarian Claim
Hegel s Trinitarian Claim G. W. F. Hegel is one of the greatest thinkers of the GreekWestern trinitarian tradition. He said that the theologians of his day had effective ly abandoned the doctrine of the
More informationP (A) = lim P (A) = N(A)/N,
1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or nondeterministic experiments. Suppose an experiment can be repeated any number of times, so that we
More information1/8. Descartes 4: The Fifth Meditation
1/8 Descartes 4: The Fifth Meditation Recap: last time we found that Descartes in the 3 rd Meditation set out to provide some grounds for thinking that God exists, grounds that would answer the charge
More informationIEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem
IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have
More informationLecture Notes, October 30. 0. Introduction to the philosophy of mind
Philosophy 110W  3: Introduction to Philosophy, Hamilton College, Fall 2007 Russell Marcus, Instructor email: rmarcus1@hamilton.edu website: http://thatmarcusfamily.org/philosophy/intro_f07/course_home.htm
More informationDate of decision 15 July 1986. Case number T 0208/843.5.1 Application number EP79300903 G06F15/20
Date of decision 15 July 1986 Case number T 0208/843.5.1 Application number EP79300903 IPC G06F15/20 Procedure Language EN Title of the application Applicant name VICOM Opponent name Headnote I. Even
More informationUnifying Epistemologies by Combining World, Description and Observer
Unifying Epistemologies by Combining World, Description and Observer Stuart Umpleby Research Program in Social and Organizational Learning The George Washington University Washington, DC Umpleby@gwu.edu
More informationIn this essay, I will first outline my understanding of the basis for Kant's categorical
I ought never to act except in such a way that I can also will that my maxim should become a universal law. What does Kant mean by this, and does it give the right kind of guidance when we are choosing
More informationThe Gospel & The Scholars. For most of us, our college days are a time in our lives centered around study, research,
1 The Gospel & The Scholars William K. Lewis Fairmont Presbyterian Church College Ministry Team For most of us, our college days are a time in our lives centered around study, research, and learning. We
More informationDRAFT TJ PROGRAM OF STUDIES: AP PSYCHOLOGY
DRAFT TJ PROGRAM OF STUDIES: AP PSYCHOLOGY COURSE DESCRIPTION AP Psychology engages students in a rigorous appraisal of many facets of our current understanding of psychology. The course is based on the
More informationThe Alignment of Common Core and ACT s College and Career Readiness System. June 2010
The Alignment of Common Core and ACT s College and Career Readiness System June 2010 ACT is an independent, notforprofit organization that provides assessment, research, information, and program management
More informationThe Basics of Graphical Models
The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures
More informationWhat is Undergraduate Education?
Education as Degrees and Certificates What is Undergraduate Education? K. P. Mohanan For many people, being educated means attending educational institutions and receiving certificates or degrees. This
More informationSkepticism about the external world & the problem of other minds
Skepticism about the external world & the problem of other minds So far in this course we have, broadly speaking, discussed two different sorts of issues: issues connected with the nature of persons (a
More informationSocial & Political Philosophy. Karl Marx (18181883) Economic and Philosophic Manuscripts of 1844
Marx 1 Karl Marx (18181883) Economic and Philosophic Manuscripts of 1844 Estranged Labor Marx lays out here his theory on the alienation of labor Marx s thesis would advance the view put forth by Rousseau
More informationBook Review of Rosenhouse, The Monty Hall Problem. Leslie Burkholder 1
Book Review of Rosenhouse, The Monty Hall Problem Leslie Burkholder 1 The Monty Hall Problem, Jason Rosenhouse, New York, Oxford University Press, 2009, xii, 195 pp, US $24.95, ISBN 9780195#67898 (Source
More informationA GENERAL CURRICULUM IN MATHEMATICS FOR COLLEGES W. L. DUREN, JR., Chairmnan, CUPM 1. A report to the Association. The Committee on the Undergraduate
A GENERAL CURRICULUM IN MATHEMATICS FOR COLLEGES W. L. DUREN, JR., Chairmnan, CUPM 1. A report to the Association. The Committee on the Undergraduate Program in Mathematics (CUPM) hereby presents to the
More informationProving God Exists. Ashley Kerner
Proving God Exists Ashley Kerner Submitted in Partial Fulfillment of the Degree of Bachelor Arts in the Integral Curriculum of Liberal Arts at Saint Mary s College April 16, 2012 Advisor: Preface During
More informationIntroduction. I. Proof of the Minor Premise ( All reality is completely intelligible )
Philosophical Proof of God: Derived from Principles in Bernard Lonergan s Insight May 2014 Robert J. Spitzer, S.J., Ph.D. Magis Center of Reason and Faith Lonergan s proof may be stated as follows: Introduction
More informationWhat Critical Means in Critical Thinking
Donald Jenner Department of Business Management Borough of Manhattan Community College The City University of New York 199 Chambers Street, Room S660 New York, NY 10007 2122208205 What Critical Means
More informationNECESSARY AND SUFFICIENT CONDITIONS
Michael Lacewing Personal identity: Physical and psychological continuity theories A FIRST DISTINCTION In order to understand what is at issue in personal identity, it is important to distinguish between
More informationIS YOUR DATA WAREHOUSE SUCCESSFUL? Developing a Data Warehouse Process that responds to the needs of the Enterprise.
IS YOUR DATA WAREHOUSE SUCCESSFUL? Developing a Data Warehouse Process that responds to the needs of the Enterprise. Peter R. Welbrock SmithHanley Consulting Group Philadelphia, PA ABSTRACT Developing
More information#HUMN104 INTRODUCTION TO PHILOSOPHY
Coffeyville Community College #HUMN104 COURSE SYLLABUS FOR INTRODUCTION TO PHILOSOPHY Mike Arpin Instructor COURSE NUMBER: HUMN104 COURSE TITLE: Introduction to Philosophy CREDIT HOURS: 3 INSTRUCTOR:
More informationLecture 9 Maher on Inductive Probability
Lecture 9 Maher on Inductive Probability Patrick Maher Scientific Thought II Spring 2010 Two concepts of probability Example You know that a coin is either twoheaded or twotailed but you have no information
More informationFall 2012 Q530. Programming for Cognitive Science
Fall 2012 Q530 Programming for Cognitive Science Aimed at little or no programming experience. Improve your confidence and skills at: Writing code. Reading code. Understand the abilities and limitations
More informationOne natural response would be to cite evidence of past mornings, and give something like the following argument:
Hume on induction Suppose you were asked to give your reasons for believing that the sun will come up tomorrow, in the form of an argument for the claim that the sun will come up tomorrow. One natural
More informationUndergraduate Psychology Major Learning Goals and Outcomes i
Undergraduate Psychology Major Learning Goals and Outcomes i Goal 1: Knowledge Base of Psychology Demonstrate familiarity with the major concepts, theoretical perspectives, empirical findings, and historical
More informationIntroduction to Proofs
Chapter 1 Introduction to Proofs 1.1 Preview of Proof This section previews many of the key ideas of proof and cites [in brackets] the sections where they are discussed thoroughly. All of these ideas are
More informationBioethics Program Program Goals and Learning Outcomes
Bioethics Program Program Goals and Learning Outcomes Program Goals 1. Students will develop a solid knowledge base in areas of Biology including cell biology, evolution, genetics, and molecular biology.
More informationStrictly speaking, all our knowledge outside mathematics consists of conjectures.
1 Strictly speaking, all our knowledge outside mathematics consists of conjectures. There are, of course, conjectures and conjectures. There are highly respectable and reliable conjectures as those expressed
More informationCSC384 Intro to Artificial Intelligence
CSC384 Intro to Artificial Intelligence What is Artificial Intelligence? What is Intelligence? Are these Intelligent? CSC384, University of Toronto 3 What is Intelligence? Webster says: The capacity to
More information