In your definition of the problem you state that "Induction is logically inconsistent in that the number of possible cases which might be observed is infinite and no number of observed cases which give the same result can therefore increase the probability of that result occurring in the next observed case, because no number can make a dent on infinity. Probability remains zero". I do not know if this is a generally accepted view or your own particular view. However I would like you to explain how you know that the number of possible cases is infinite. For instance we believe that based on scientific enquiry Fluoride in drinking water arrests tooth decay although one day some unfortunate person may rapidly loose all his teeth from exposure to fluoride. Now there are only a finite number of persons living in the world and who have lived in the world at any one time and notwithstanding how many births there are after that time there will still only ever be a finite number of subjects for observation. How does this relate to your suggestion of an infinity of cases?

I am having difficulty in understanding that what you have said is a solution to the problem of induction. We cannot get out of our heads, minds, or what you will to examine the world in its raw state and I do not suppose we would understand it if we could. We only know the world by our perceptions obtained in the main by virtue of the senses governed by the limitations of them and the limitation of our ability to interpret. In view of these limitations we can only proceed say with scientific enquiry by virtue of the inductive process. This entails looking at beginnings and ends and bits and pieces we extract from what we believe is reality and even before doing this we have found it necessary to assume 1. That nature is uniform and 2. that every event has a cause. This is the only way given to humans to make enquiries of this nature and I believe most who have considered the problem to be aware of this. Induction does not give certainty only sound arguments and probabilities. To solve the problem of induction is surely to get around these profound difficulties in order that we may with certainty understand the World but it seems to me that only God if he exists is in this enviable position.

Eric Replies: [Return to Reaction and Comments page] or [Return to Home Page] or [Top of Page]

With regards to infinite cases, I meant that the discrete "events" which might be mentally created are infinite, because we can mentally divide up the process of reality in an infinite variety of ways. In addition, each individual observer occupies a unique point in time and space and it is only by a mental abstraction resulting in a generalisation that comparison between disparate "events" is possible - but the logical validity of such generalisation is exactly the point at issue. However, your point is a good one, and one which troubled me before I actually wrote the article. Perhaps it is a point which does not lend much support to the main theme. But the view is not in fact my own, it is general amongst philosophers, (in so far as explaining what the problem of induction is, I mean). If we say that cases are not infinite, at what point is certainty (or probability) to be achieved? After all, we do not know the number of cases in question, and therefore we cannot make any mathematical calculation in dealing with the matter. So how many cases must be observed before a general claim can be justified? How logical is that justification? Even if we leave infinity out of the case, these questions still demand answers.

The "solution" I offer, by the way, is simply a proposal. If we are aware of the limitations of what we hold as knowledge, we may gain a greater degree of flexibility in our examination of the world as we perceive it. If we realise that the useful "logical fictions" we employ are not realities, but simply the mental tools we use to examine and manipulate reality, then we may perhaps choose and examine a little more carefully and have a little more respect for the process we examine and experience. I was not saying that Reality does not have repeating patterns, I was rather examining the logical principles upon which the expectation of such repetition is based. I agree that there are such repeats. But philosophy often questions the obvious, in order to better understand the assumptions upon which everything is based. It appears to me that the assumption we have for repeats is built in to us by evolution. Because we are part of Reality, part of Nature, this assumption has helped to ensure our survival, and therefore gives the ultimate proof of the reality of pattern.

However, you do say that every event has a cause? Consider this: when the cause of something is discerned, do we not discover the cause by working backwards from the effect? And is not the decision that a thing is the cause defined by the effect for which we seek a cause? If this is so, then cause and effect are discovered by the human observer who seeks them out, according to his definition of the effect and the cause.

If we seek the cause of the gap between a lightning flash and a rumble of thunder, we might say that the thunder is the sound of the lightning crack, and because light travels faster than sound, this is the cause of the gap between the lightning flash and the thunder's rumble. But in reality, both are simultaneous. We could equally say, the cause of the gap between the lightning we see and the thunder which follows, is rather attributable to the distance we are from the source of the storm. Which is the cause of the gap? Surely it is both? The cause to be sought is defined by an observed effect and the observer is a human being with the assumptions, needs, desires, wants and failings of a human being. There is nothing in nature which gives absoluteness, save the best efforts of human seekers to be objective and precise. It is all relative to the Human. Thus we gain a knowledge which can never be perfect, for those that come after us may always discover a better, more truthful description of the causes and the effects we observe. Some effects, then, are to do with a relationship to a cause, not just to a cause alone.

James Haryett writes: [Return to Reaction and Comments page] or [Return to Home Page] or [Return to Top of Page]

I don't think that Don Birds argument is correct since in his example he states that "there are only a finite number of persons living in the world". Now this is true that if you could freeze-frame time in one moment there will be finite number of people to count and if you wanted to do a survey of the effects of fluoride on this very large group of people then you could come to some conclusion. The problem is that you have now put your own limitations on the study, you've created a beginning and an end. But who's to say that there isn’t a genetic sequence in dna which will in fact react with fluoride and breakdown, but that sequence has not been formed yet. How do you know that this one person (with the anti-fluoride dna) who will no doubt pass on their dna was just not in your initial group (they haven’t been born yet). So you then wait 1 year and then freeze time again, then 1 year later again. Now you would no doubt have new people in each group to study but now the study is infinite because where can you stop? If you stopped after 50 years and this one person born with the anti-fluoride dna sequence was born 51 years after you started the study, then your inductive conclusion is wrong! Therefore the study groups would have to be infinite because there is no telling where to end. Therein lies your problem with inductive logic.

Don Bird Writes: [Return to Reaction and Comments page] or [Home Page] or [Top of Page]

Mr Harryett’s comments; he mentions freeze framing and putting temporal limits on the study. I see no other way to examine the effects of fluoridated water on Human beings. To start with even with a finite number of people in the world we obviously cannot examine them all so we must be selective. Initially from observations made an hypothesis has been formulated that “fluoridated water reduces tooth decay.” Accordingly with as many subjects as can be accommodated, and/or thought necessary, one sets about verifying this hypothesis by controlled experiment and statistical analysis of the results. Out of this it is found that those exposed to fluoridated water suffer less tooth decay than those exposed to non-fluoridated water. This leaves a verified hypothesis. But of course this is not the end, the final irrevocable answer, as all science is tentative, and likely to be revised in the light of further knowledge. It will never be certain that the verified hypothesis will hold for all time. There may well be genetic mutations and other factors conspiring against the hypothesis and that is why vigilance is of prime importance in all scientific work. Mr Haryett states the study group would have to be infinite. Well this is a philosophical conclusion, I would prefer to say that watchfulness, observation, alertness and caution should prevail continuously as the result of induction is at best probability never certainty.

The postulate of induction that Nature is uniform, presents a substantial problem. Evidently Nature is not uniform, there are for instance, meteorological variations, a history of Earth magnetic field reversal, gravity anomalies, continental drift, genetic mutations to name but a few. Replications of instances are not possible all we can hope for is similarity, resemblance between the past and the future, but never duplication. So far as we know the total state of the universe has never repeated itself. So only roughly speaking, is there a measure of uniformity in nature. Further problems are also revealed in the sceptical approach. If the sun fails to rise on any particular day then we may think our background assumptions leading us to expect it would, are wrong, and certainly if we live in the polar regions there are periods when the sun does not rise each day. However if the sun does rise scepticism still prevails, as there is no foolproof way of ascertaining that it rose for the same reason as before. It seems that on the face of it our spatio-temporal experience, which seems consistent, could be a poor guide to understanding reality. Here our so-called consistency seems also based on superstition, we have to fall back on supposition, habit, condition of the mind and unreflective habits of grouping together experiences. Nelson Goodman (Fact Fiction and Forecast iv) has described how we make judgements between competing hypotheses concerning which one seems to embrace the greatest explanatory and predictive success, and rely on entrenched ideas in our explanations. Against such a background together with all the vicissitudes of human nature besetting the observer we ask, or expect, if like antecedents recur then like consequents will or should follow. It is accordingly somewhat difficult to see how induction can be vindicated or justified.

The problem of induction is a philosophical problem not a scientific one. Science makes great and apparently successful progress in the face of it, or perhaps oblivious to the problem, but never ignoring the fact that scientific progress is tentative. C. D. Broad said that induction “ is the glory of science and the scandal of philosophy.” But is it a scandal? Such a description can only arise from a refusal to accept induction for what it is; a system, which by and large gives reliable answers but has an element of untrustworthiness built in to it. It cannot be justified; locked away from reality by our own cognitive limitations, perhaps we are merely looking at the menu and thinking it is the meal, but induction is the only way open for us to make some sense of the world, and can only be used in accordance with good human reasoning and judgement.

Don Bird Writes (some years later): [Return to Reaction and Comments page] or [Home Page] or [Top of Page]

I am currently writing an essay on the problems of Induction, and came across your solution with which I agree. I have always suspected this is a pseudo problem and Hume's efforts to find a necessary connection between cause and effect were doomed to failure because there is none. His psychological answer that it was merely habit, a condition of the mind, which bridged the gap in inductive reasoning does seem in the light of your solution, the only conclusion he could have reached. To speak of cause and effect is unjustifiably to single out two events from what is in fact a continuous series. Your solution reminded me of the Work of Alfred North Whitehead who to put it crudely saw reality as an organism a continuous process. We do examine, in what Whitehead called simple location, bits of this and that and make pronouncements like A causes B and this is all well and good for our purposes of getting through life. By such methods humans can, say, discover new drugs to treat illness. However the great error occurs when we interpret reality in accordance with what we find in these bits. This according to Whitehead is to commit the Fallacy of Misplaced Concreteness. I was wondering if you had been influenced by Whitehead, he is obscure and almost impossible at times as you probably know and your reference to the Aristotelian solution to Zeno is in some ways similar; I liked the idea of a line having only the potential for infinite division.

Eric Replies: [Return to Reaction and Comments page] or [Return to Home Page] or [Top of Page]

I’m pleased that, having gone further into philosophy, you have come to agree with my proposed solution to the problem of induction and that you can still find the website interesting. I have not read Whitehead’s works, but I’m familiar with some of his main ideas through summaries and definitions, etc. I was not aware of any great Whitehead influence on me, although one can never be absolutely certain when one reads continuously, as I did when younger. I have read a lot of Russell, who does mention Whitehead’s work very occasionally. So far, the only influences that have been indicated by site visitors are those of Popper and Kant, both of which I acknowledge (Kant as major, Popper as minor). Whitehead comes as a bit of a surprise, yet often the ideas of philosophers “filter down” into the background mentality of an individual or even into the background of a whole culture, coming to be accepted without attribution. Certainly, your exposition of Whitehead’s ideas does show them to be pertinent when compared with the points I make in dealing with the problem of induction in my website article.

Don Bird Writes: [Return to Reaction and Comments page] or [Return to Home Page] or [Top of Page]

So far as I can see Whitehead did not actually set out to demolish Induction. The system he proposed, sometimes called Process philosophy leaves little room for the ideas of cause and effect and induction as we generally understand them, but I think he would agree that science cannot proceed without abstractions and the idea of simple location. His point of view however was not to banish these from scientific practice but merely to convey that our understanding of the world entailed this approach. This is the manner in which we make sense of the world but it does not represent reality which, as I think you will agree, is a process and not a collection of events. Whitehead's ideas are intrinsically difficult to grasp. They involve a total reconstruction of our concepts as to the nature of the universe. Even when persuaded by his arguments on an intellectual level, one may still be left in the position of failing to summon to the imagination exactly how the reality he describes is structured and how it can exist at all. The usual notions of place, time, and things are abandoned. We are asked to see the universe as a process neither physical nor mental but something like a "feeling". This process we are told goes on everywhere all of the time. I include here a quotation from page 51 of Whitehead's Science and the modern World.

I would not worry that Whitehead may have pre-empted you in this matter I think you could console your self that you are lucid and he certainly is not, which probably accounts for the fact that he is currently little read. When asked once why he did not write more clearly he replied "because I do not think more clearly" I must admit I spent many hours trying to ascertain whether he did or did not find a place for causation and induction in his philosophy of organism; if he did then it is distorted out of all recognition. I feel pretty sure that your analysis of the problem is as I said, lucid and most probably correct.

Eric Replies: [Return to Reaction and Comments page] or [Return to Home Page] or [Top of Page]

Thanks for your comforting words. If Whitehead did not apply his ideas to the area I am interested in, and did not develop them with regard to Darwin, there’s still a chance I may have produced something original, even if it is an amalgam of already proposed theories.

I have altered my website's Home page to give an acknowledgement to Whitehead. Thanks for your help in this matter.

Tim Powys-Lybbe Writes: [Return to Reaction and Comments page] or [Return to Home Page] or [Top of Page]

It appears to me that you are constraining the problem of induction to probabilities of perception. If something has been perceived before, then how likely is it to be perceived again. Not very is indeed the correct answer.

What you are trying to do is to define a scientific theory in terms of human perception. Popper made a valiant attempt to do the same and said that a theory was scientific if it predicted new observations. This is fine when people are uttering new theories but becomes very difficult to handle for old ones. One has to ask the historical question of whether the theory predicted new events at the time it was first proposed.

I do not think that any account of the logic of scientific theories will work if it is primarily in terms of what was perceived, the fleeting events that pass one's eyes.

If, on the other hand, we were to ask the average scientist what they thought they were doing, they would say that they were trying to find out how the world worked. The principle of induction they use is not that a similar perception will occur again, or that a new perception will once upon a time have occurred. Rather they are saying that the real world we know behaves in fixed ways and that in similar circumstances will behave similarly; allowing for statistical as well as mechanistic descriptions of course.

If it is part, as they would say, of something's nature to behave in a particular way, then that is what they are trying to discover, the laws of nature. As we know only too well, the discoveries are initially imprecise and halting but as the years progress some things become more and more accepted as definitely part of nature and others are heuristic, a developing understanding of what the natural world might be.

I am, of course, assuming that there is a world out there, beyond my perceptions...

Eric Replies: [Return to Reaction and Comments page] or [Return to Home Page] or [Return to Top of Page]

Your comments are appreciated. It would seem, however, that you have taken a part of the argument concerned with stating the problem (concerned with clarifying and defining it) and have perhaps not realised that my proposed solution is suggesting more or less what you are saying in the second half of your message.

In the solution proposed, I suggest that we discover trends in the fabric of reality, by examining the weave, the threads of causality we can observe, which we pick out through an expectation of pattern. The discovery of trends in reality is based upon not only observation, but on an inescapable "build-in-by-evolution" assumption of pattern. This is where the predictive force arises from: from an expectation of pattern which is a part of the structure of the brain, developed over millions of years of interaction with reality. It is the forces of nature which have built in the expectation, and it has spread and developed because (all things being equal) it works - it is an expectation that helps (generally speaking) to ensure the survival of the creature having the expectation. This is the same expectation which gives Logic its force. Without this expectation, there could be no Logic. This expectation expresses the coherence of reality.

I hope that this makes it clear that although I am constraining the problem of induction to probabilities of perception in stating the problem itself, so that it may be understood by those unfamiliar with it, I am not constraining it thus in my proposed solution (not in the sense you indicate, as an event which passes before the eye). That part of my article was simply an attempt to show how the problem arises, byway of such constraint. My proposal is that there is an important mental element which has been either overlooked or underestimated in its influence. The event itself is defined as an Event, which is a mental invention. What is happening, in my opinion, is not simply the fleeting events which pass before the eye, but the observation of reality (a process), plus what happens within the brain afterwards to produce a logical fiction which we can work with.

Popper was seeking to define a scientific methodology which would clarify a demarcation between knowledge and belief. He wanted to show that scientifically obtained knowledge was primarily based upon the application of logic and therefore predictive force arose from the falsification of predicates and the testing of predictions. I am saying that the acquisition of all knowledge arises from a long time interaction between malleable living things and reality. It is reality which has programmed in the ability to be predictive concerning reality. Science is a more refined and more universally checked occurrence of this, which has greater persuasive and predictive power because it claims the discovery of pattern with greater logical clarity through the strictness of its definitions and classifications.

Jon Sellars writes: [Return to Reaction and Comments page] or [Return to Home Page] or [Return to Top of Page]

The stadium, it tends at first view to hold true that one cannot travel an infinite distance in a finite time but then one considers that we are talking of a finite distance since we have specified this originally and thus a finite distance can be travelled in a finite time, this distance does indeed have the potential to be infinitely divisible. One would then say the sum of the infinite amount of divisions will equal a finite distance. The argument is not the same as if one asked if I set of from earth how long would it take to come to the end. In this case one could argue that the distance one can travel is infinite since no point as been set for termination and therefore one would travel infinitely.

Eric Replies: [Return to Reaction and Comments page] or [Return to Home Page] or [Return to Top of Page]

The paradox is concerned with saying that anything travelling any distance (say, one mile) must pass through an infinity of points. This is because numbers are infinite, and therefore we can mentally divide whatever distance remains using number. Before we can travel the one mile, in other words, we must travel a half of a mile; before we can travel half a mile, we must travel a quarter of a mile; before we can travel a quarter of a mile, we must travel an eighth of a mile…and so on, infinitely. We can always say, even of the tiniest movement, that half of such a movement could be calculated. So, according to Zeno’s paradox, movement becomes logically impossible, even over a finite distance (one mile), seeing as it encompasses passing through an infinite number of points. It is not the distance itself which is infinite to our minds, but the number of divisions which lie within that distance. Zeno is arguing that this paradox reveals to us that the evidence of the senses is faulty, and that the illusory nature of reality is revealed by logic, thus showing logic to be superior to the senses in gaining access to the truth.

I hope that this recap helps to fix the implications of the paradox in your mind.

You are correct when you say that this is not the same thing as saying if I set of from earth how long would it take to come to the end. However, this statement assumes an end, and in assuming an end, we place a tacit termination to the movement. This is the way the human mind works. We mentally create Events, not processes. We could still say that we will reach a point which is half way to the end. The only way around this is to claim that the “End” in question is constantly travelling away from us (in other words, infinity). If we say that we are travelling towards a point which is constantly getting further away from us (as in, say, an expanding universe), we can never reach any halfway point. If we can never reach halfway, we cannot calculate halfway and therefore Zeno’s logic breaks down (this is a variation on Aristotle’s counter to the paradox, which involves an examination of the nature of infinity).

My own counter to Zeno is somewhat different. I claim that the human mind often works (thinks about reality) byway of Logical Fictions. This counter examines the ontological nature of such things as Number and Events. I am asking what such things really are. I find that they are mental in nature.

Jon Sellars writes: [Return to Reaction and Comments page] or [Return to Home Page] or [Return to Top of Page]

I was thinking about the thing with the swans and it would seem that one could say that it is likely considering past events that a white swan will fly past next or one could not consider past events and say that either a black or white swan could fly past next. The thing I would most likely do is to join the two statements together and say that it is probable that a white swan will fly past next but not improbable that a black one will.

Eric Replies: [Return to Reaction and Comments page] or [Return to Home Page] or [Return to Top of Page]

I think your final statement, combining the two previous conclusions, attempts to cover all possibilities, rather like the statement: “Either the sun will rise tomorrow or it will not”. The trouble with such statements is that they are, in effect, claiming nothing in the way of knowledge. But the problem of induction revolves around a definition of true Knowledge, what it is and how we can acquire it. Your proposed solution also raises a host of new problems concerning the meaning and definition of such terms as “probable” and “improbable”. How many white swans observed make it probable that the next one observed will be white? How improbable does it become, that the next swan observed will be white, if one black swan has just be observed?

In reality, we are not so discriminatory. I am more interested in the way we think, not so much in the way we should think (I don’t think that it is possible to proscribe in this area, anyway).

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