A new cure for insomnia

Just so you don’t think it’s all work and no play around here, I wanted to show you how we statisticians really unwind and let go. We write papers that as many as ten or even twenty other guys around the whole world will read.

My friend Russ Zaretzki and I just submitted the paper “Induction and Falsifiability in Statistics” to Bayesian Analysis.

The abstract:

The importance—and rationality—of inductive arguments and their relation to the frequently invoked, but widely and poorly misunderstood, notion of falsifiability are explained in the context of statistical models. We remind readers that no probability model can be falsified. Both frequentists and Bayesians must use inductive arguments. This includes arguments for the use of p-values and those given in model selection and for the creation of goodness of fit measures. Since only Bayesian theory is equipped to put probabilities on the conclusions of inductive arguments, we argue that even frequentists are Bayesians at heart.

Since we’re interested in how peer-review works, I’ll try and keep us posted as this paper makes its way through the process.

A snippet from the Introduction (I’ll leave the bibtex references alone and opaque):

Regardless of the common sense of (2), the early part of the 20th century saw the growth and dispersal of the belief that all inductive arguments are unreasonable. The philosopher most responsible for this view was Karl Popper \citep{ThePsi1987,GroLev1994}. Popper asked, “Are we rationally justified in reasoning from repeated instances of which we have experience [like the hot flames] to instances of which we have had no experience [this flame]?” His answer: “No” \citep{Pop1959}. This extreme skepticism has understandably not been accepted by many philosophers \citep{Car1950,Haa2003,ThePsi1987,Wil1947,Sto1982}, but as we shall see it has been, at least in some form, by statisticians and probabilists.

Popper convinced many that since induction could not and should not be trusted—because it might lead to an invalid conclusion—only deduction should be used in scientific inference. Since it is difficult to prove things deductively, Popper therefore claimed that the mark of a scientific theory is that it can be falsified; theories that could not be were said to be metaphysical or not scientific. Now, the term falsified has a precise, unambiguous, logical meaning: that something was shown to be certainly false. Despite this simple definition, there have developed many odd, and incorrect, interpretations of this word in our community, which we detail below.

First, the falsifiability criterion is obviously useless for theories that are true (such as in math) and therefore cannot be falsified. Falsifiability is also useless with statistical arguments. This is because they use probability statements which cannot be falsified, and therefore are, in Popper’s scheme, metaphysical.

No model or theory that makes a probability statement (between 0 and 1) can be falsified because there can exist no set of observations which are logically inconsistent with any probability statement. An example, “This logistic regression model says the probability of rain tomorrow is 0.9.” Either observation, rain or now, is logically consistent with that statement. It cannot be falsified.

What fun!

39 Comments

  1. Promises to be an excellent read. Thanks.

    BTW, was it only mr Russ that wrote that article, or you are also a de facto co-author? You don’t mention yourself at all in your post, but your name is the first one in the paper.

  2. I’m definitely interested how THIS peer review will go. I have a hunch it will be different than virology peer review, where the biggest hurdle tends to be competition for space in the journals. Many times the response we get from the reviewers is something like

    “this manuscript has no glaring flaws and would be of publication quality, however due to a limited number of articles per issue, we feel the topic is too specific and thus outside the scope of this journal. We suggest resubmission to a more specialized journal where your topic will appeal to a larger percentage of the readers.”

    In such a case our original submission may be to PNAS and resubmision goes to ‘Journal of Virology’ or ‘Emerging Infectious Disease’ without problems, probably by he same reviewers and a different editor.

  3. Very interesting indeed. I can’t read the algebra of it, but I appreciate the philosophical writing and insight.

    At first I thought “Hume” was David Hume, who, after all, made some interesting remarks at the probabilities of miracles :). But then I saw a reference to Hume 2003, which after all is a miracle of itself, and found myself corrected!

  4. Good stuff. Could’ve used a little more exposition on the finishing paragraph rather than a smattering for symbols. I give high marks for actually translating your symbolic statements as well. I have many “students” who can’t go from math –> words and vice versa. Let’s just say that where I went to university they used an entire class on propositional calculus, logic, and proofs for their “flunk-out” class.

  5. speaking of Zaretzki, what happened with the paper you mentioned about ” probability depends on knowing the future number of trials ” ?

  6. “First, the falsifiability criterion is obviously useless for theories that are true (such as in math) and therefore cannot be falsified”

    This is a fundamental misunderstanding of the term, and one so elementary that I have trouble believing that anyone in the field would write it.

    To be falisfiable, a theory must be so defined that it is possible to put it to a test that would falsify it. Nothing more. If I say that God created the universe yesterday, and all of our memories of earlier life are false, that would not be a scientific claim. In spite of the fact that it could be true, we cannot prove it is not true – God’s supernatural powers would always be left as a possibility.

    You are confusing a scientific theory – a hypothesis that has been supported with scientific evidence over time – with the facts of the physical world. Biological evolution through natural selection is a scientific theory that explains the living world The evolution that has gone on through the ages is the physical fact that has been discovered through the development of the theory. Every accepted scientific theory should begin with a hypothesis, and then be examined and subject to falsification. When falsification fails, then the hypothesis is provisionally accepted. Still, we should still accept that if new information comes along, a theory that appears to be true now could be falsified at a later date. It happens. All the time. So your “theories that are true” is actually an unscientific one. It suggests that we know when science stops, like the guy who wrote the book about “The End of History.” His end of history lasted for about five years.

  7. Mark,

    Nope. I’m afraid you’re used to the popularizations of Popper’s views. I gather, also, that you did not read past the Introduction to the details about probability models and falsification.

    Let me just ask you this: take any probability model—or theory—that makes a prediction. To make that concrete, make it a normal probability model (as is every linear regression etc.). Now you tell me how you can “put it to a test that would falsify it.”

    I’ll wait here.

    Update (sorry, am in a big hurry this afternoon): I should mention I am far from the first to make the criticisms about falsifiability. Here’s another one you forgot. Take any argument/theory with true premises and a valid conclusion. Tell me how you can “put it to a test that would falsify it.”

    I’ll still be here.

  8. A. “First, the falsifiability criterion is obviously useless for theories that are true (such as in math) and therefore cannot be falsified”

    B. “Take any argument/theory with true premises and a valid conclusion.”

    Mathematical proofs are nevertheless based on premises, so even if the conclusion is completely valid, there is still the possibility that the premises may be found wanting.

  9. Godel proved that maths aren’t “true” by self-definition. They aren’t “contained”, so to speak. Maths depend on axioms that are taken for granted. That is, we consider them to be so, inductively.

    I also have problems with this sentence that has been quoted by MarkB, I’ll repeat it because everyone seems to like it :):

    “First, the falsifiability criterion is obviously useless for theories that are true (such as in math) and therefore cannot be falsified”

    How can you claim that math is “true”? Or any theory that is “true”? I’m having problems with this definition. Perhaps you meant “true until proven otherwise“. And if they are true, they can always present plausible tests for its own falsifiability.

    Example. Theory of Evolution. “True”. Can be falsified, for instance, if you detect rabbit fossils in the Cambrian (how do they say it?) plate (?). Well, sure, it may not be entirely falsified, but it does count as an heavy blow.

    Example. Theory of Gravity. “True”. Can be falsified, for instance, if something that ought to fall, and while any other force is absent, doesn’t. It is falsified if the calculations done by Galileo and Newton aren’t met by empirical evidence. Etc.

    Or, you meant “True” as in “We take these as Given truths, as Self-evident Truths”. Well, if you take them as granted, then surely you aren’t interested in falsify them, now are you!!

  10. “Take any argument/theory with true premises and a valid conclusion. Tell me how you can “put it to a test that would falsify it.”

    an argument with true premises will ultimately contain a conclusion that would ‘appear ‘
    valid until it has been proven false.

    Newtonian physics has lots of “true” theories and premises with valid conclusions which ultimately have been shown to be false, yet very close approximations (when dealing with normal eath sizes and speeds) of other “true” theories premises and formulas corrected by einstein/others, which will likely be proven false again with minor corrections for variables we’ve not yet considered.

  11. at the risk of spamming your blog, I will also take a stance against this:

    “No model or theory that makes a probability statement (between 0
    and 1) can be falsi fied because there can exist no set of observations
    which are logically inconsistent with any probability statement.”

    I agree you cannot falsify the conclusions of such a model or theory, but you can certainly falsify the model or hypothesis if the declarative statements which lead to the probability conclusions are shown to be wrong.

  12. Briggs: “Popper convinced many that since induction could not and should not be trusted—because it might lead to an invalid conclusion—only deduction should be used in scientific inference. Since it is difficult to prove things deductively, Popper therefore claimed that the mark of a scientific theory is that it can be falsified; theories that could not be were said to be metaphysical or not scientific. Now, the term falsified has a precise, unambiguous, logical meaning: that something was shown to be certainly false. Despite this simple definition, there have developed many odd, and incorrect, interpretations of this word in our community, which we detail below.”

    PG: Actually Popper used “falsifiable” and “testable” as equivalent terms throughtout his writing.

    Popper: “A theory is tested not merely by applying it, or by trying it out, but by applying it to very special cases — cases for which it yields results different from those we should have expected without that theory, or in the light of other theories.” [Conjectures and Refutations]

    Briggs: “First, the falsifiability criterion is obviously useless for theories that are true (such as in math) and therefore cannot be falsified. Falsifiability is also useless with statistical arguments. This is because they use probability statements which cannot be falsified, and therefore are, in Popper’s scheme, metaphysical.”

    PG: Mathematical theories are quite different to scientific theories. Nevertheless, if mathematical theories were untestable, they would not be accepted as true. Probability statements can most certainly be tested and falsified. Modern particle physics is chock-a-block with them. The suite of theories called Quantum Mechanics is widely believed to be the most corroborated physical theories of all time.

    The theories of which Popper wrote were:

    1. Logical & mathematical thories
    2. Emprical & scientific theories
    3. Philosophical or metaphysical theories.

    “How can we, in each of these groups, distinguish between true and false theories? Regarding the first group, the answer is obvious. Whenever we find a mathematical theory of which we do not know whether it is true or false we test it… by trying to refute it… to prove it or refute its negation.

    ….

    If we now look at emprical science, we find that we follow, as a rule, fundamentally the same procedure… The important difference is that now we can also make use of empirical arguments.”

    Now if we try to use these methods on metaphysical theories we have a problem. Or rather, a set of problems many of which have failed to succumb to testability for thousands of years. Consider the following:

    Idealism: the world is but a dream
    Determinism: the future is entirely a result of the present
    Indeterminism: events in the world occur purely by chance

    …and so on. All of these theories are irrefutable.

    Briggs: “No model or theory that makes a probability statement (between 0 and 1) can be falsified because there can exist no set of observations which are logically inconsistent with any probability statement. An example, “This logistic regression model says the probability of rain tomorrow is 0.9.” Either observation, rain or now, is logically consistent with that statement. It cannot be falsified.”

    PG: However, taking “this logistic regression model” to be the one Mike Pook was using to predict the probability of rainfall, temperature and wind, then it was certainly tested and found to be accurate 70% of the time. I’d say it was tested and sufficiently corroborated to be useful. So would Popper.

  13. All,

    I encourage everybody to read the entire paper, at least the quotes of Popper. It will be fun to go over those. They sound pretty irrational, do they not?

    Most people, if they have a standard science education, have only heard of Popper’s falsification ideas in a positive light. They will not be familiar with its standard critiques, which I again assure you are not unique with me. This is usually because people are taught Popper’s ideas by scientists who themselves are not aware of its shortcomings. (This happens in statistics all the time, when psychology or sociology professors teach statistics, none of them being familiar with any criticisms/failings of the classical theory.)

    Also please read the references given in the Introduction. There are many others relevant works which I did not cite in the paper but I’ll try and post here.

    All again,

    To say, “A theory is true until proven otherwise” is nonsensical. If the theory is true, it is true. End of story. Something false can be believed to be true, based on mistakes in evidence, improper premises, invalid reasoning, or other mistakes. Usually, however, something is only probably true (or false). This is so for the theory of evolution or theory of gravity, etc. I can see, however, that we will have to go over this in much more detail.

    Luis,

    I will agree with you in a way about axioms. They are not believed inductively (re-read the definition of inductive), but they are believed on faith. That is, there is and can be no evidence for their belief. We have talked about this before (search for “faith” on this blog), but we should talk about it again because it is apt to be confusing. This has fascinating and deep relevance to many areas, including the rationality of inductive arguments—and on your favorite subject…

    If an axiom is true based on no evidence—except by faith or inuition or similar word—then they cannot be proven false. So we have here examples of non-falisifiable statements that are just true. No empirical observation or additional (noncontradictory) premise can prove an axiom false.

    OMS,

    You might care to reread the second challenge sentence and ask B again.

    John,

    Your second (brief) comment is false, or rather, nonsensical. Try reading it out loud and pausing after the first three words.

    Your first examples are also false. If an argument has true premises and a valid conclusion then it is true. End of story. Nothing will come along and allow you to say it is false.

    Your third example is more plausible. I mean, it’s easy enough to stick in a premise to a probability model that you know is false, or is likely to be false. Then you can legitimately falsify the conclusion. Worse still, this happens all the time. Suppose you were to, as many people do, model systolic blood pressure (SBP) as a normal distribution. That is a premise with you. But that premise is certainly false given the premise (or observation) that SBP cannot fall below 0. The normal disitrbution will give conclusions that state the probability of SBP 0) is not part of your theory/model. Pause and absord that. The only model under consideration is the normal theory/model. Everybody with me so far? Point is: if you change, add, or subtract premisses from this model it is no longer the same model/theory. It is a new model/theory. Agreed?

    The conclusions of this original theory will absolutely give positive probability for BP values 0. Now, carefully think about this question: what observation will falsify our model/theory? We already know of an additional premiss that says our first premiss is false, but we’re not asking about new premisses, we’re asking about observations. What exact, specific observation will show our model is false?

    None. It’s true the theory will give probabilty for values < 0 and greater than, say, 300, but just because we say something is probable and we do not see that thing, it does not mean the theory has been falsified. You can observe BPs until the trump of doom and never find one that will falsify the model/theory.

    I’m running out of steam and time right now, so I’ll have to return to this. We already have a new premise that says our model is false. But we can add another that says, “It’s false, but it’s a reasonable or useful approximation.” Then nothing (well, almost nothing) will falsify that model.

    Mr Git,

    I’ll have to come back to some of this (right now, it’s off to the brewery!). I don’t accept Popper’s false distinctions between theories. Theories are theories, which are just fancy propositions. Besides, he had an explcit definition of “scientific” which does not apply to any model which is inductive, which of course is all probability

  14. John,

    What happened was the old, old story. We were scooped. So we’re turning the paper into an “educational” one which has easier proofs. It’s done: I’m waiting on Zaretzki to submit. Stay tuned.

  15. Have you have read Karl Popper’s revision of the “distinction . . . between objective and subjective interpertations of the probablilty calculus” as shown in “Realism and the Aim of Science,” ed. by W. W. Baartley, HI, (London: Routledge, 1992, 1981, pp.281-300, pp. 301, ‘Criticism of Probabilistic Induction,’ pp. 301-346, and ‘Remarks on the Objective Theory of Probability” ? He introduces a “new objective interpretation probability” which he identifies as “‘propensity interpretation.'”

    Willi

  16. Briggs,

    Since I had said essentially the same thing as Luis regarding premises and mathematical proof, the thrust of your reply is somewhat unclear to me.

    Purely mathematical theorems are accepted as “true” under a given set of premises (axioms) because the alternatives have been (exhaustively) subjected to falsification. Equivalently, the statement of the theorem itself was, in principle, falsifiable; it just did not happen to be false.

    In your reply to Luis, you say about axioms:
    “…there is and can be no evidence for their belief…If an axiom is true based on no evidence—except by faith or inuition or similar word—then they cannot be proven false…No empirical observation or additional (noncontradictory) premise can prove an axiom false.”

    The axioms themselves may be found inconsistent or insufficient through their consequences (or lack thereof). In this way, sets of axioms may be “falsified” (although the word is awkward in this usage). Even though they are intuitive, Euclid’s axioms are demonstrably inconsistent on a surface with curvature (once one wishes to explore this type of surface). One might then say that the empirical observation of curved geometries rendered this set of axioms false (even without proving another set of axioms “true”).

    Meanwhile, I have downloaded your paper and look forward to reading it!

    Cheers,

    OMS

  17. Penultimate sentence: TYPO?

    “Either observation, rain or now….”???

    shouldn’t that be : rain or not

  18. All,

    Just back from Brooklyn Brewery. Highly—and I mean highly—recommend the EIPA.

    OMS,

    Don’t forget that there are a set of axioms that lead to a consistent and complete theory of arithmetic. Move beyond, and things get dicier. One interpretation of math beyond the basics is that you need to adopt more axioms. Well, we’ll talk about that later.

    Willii,

    No worries, brother.

    Anne,

    Typo it is, sister. Regular readers know that me and orthography are not the best of friends. Thanks!

  19. It seems this paper is directed as reply to a cultural phenomena among statisticians that involves rejecting inductive reasoning because non-deductive reasoning can lead to errors. I can’t comment on that because I have no knowledge of the phenomena.

    However, your article makes the following explicit argument:

    “We merely want to show that the reasoning behind most statistical methods, and certainly those of model selection, is inductive. We thus show that falsifiability is of little or no use.

    I’ll certainly agree with the first sentence (and so I don’t need to be convinced of it). I think it’s the second sentence is a problem and I think the problem comes from your definition of falsifiable.

    Presumably a statement is falsifiable if it can be falsified where,

    “Now, the term falsified has a precise, unambiguous, logical meaning: that something was shown to be certainly false.”

    As a matter of logic you are correct that the term falsifiable as you define it is of little use because no statement that is true can be falsified. Insofar as science is concerned with sifting the true statements about the world from the false ones, the falsifiable statements would (by your definition) just tell us what NOT to believe. You present some quotes of people who seem to think this is a goal. But this is not how a scientist uses the term falsifiable. Perhaps it is how it is used in statistics but certainly not in physics and engineering.

    PG points out that Popper used falsifiable and testable interchangeably. I’m not expert or even well read on Popper but that sounds about right.

    Later in your paper you introduce the term “practically falsified” which seems closer to how a scientist uses the term.

    Here is why I do not accept your definition of falsifiable:

    The statement, “Light has the properties of a wave,” is a true statement that is falsifiable by a physicist’s understanding. The evidence that, “Light has the properties of a wave,” is a true is the following: 1) Pull hair for your head; 2) Shine a laser pointer through the hair; 3) Observe the light pattern cast on a wall a few meters away; 4) It is a diffraction pattern consistent with a wave theory of light. Note that the demonstration is not logically air tight but that is not germane. The proof that the statement is falsifiable is that there is an alternate outcome possible if light is not a wave, the laser could have created a simple shadow. The statement’s logical negation is a false statement that is falsifiable. To a physicist, that is a demonstration that the set of falsifiable statements is not a subset of false statements and the set of falsifiable statements are a subset of the sets of neither true nor false statements.

    You may say that my definition of falsifiable is based on a misunderstanding of Popper but that doesn’t matter. The definition you present for falsifiable may be logically consistent with the use of English but it doesn’t seem very useful so why would I use it?

    As I said above your, paper seems to be directed at a group that dislikes empiricism but embraces falsifiability as you define it. If that is true I certainly think that you are right and the targets of your argument are wrong. This is because induction is very useful and falsifiability as you define it is not very useful.

    I don’t know the statistics culture so I’ll throw this out. Maybe the problem is that statistics lives in the worlds of math and science simultaneously. What constitutes a convincing argument (by definition, a proof) in those two areas is not the same. The statisticians who embrace the mathematical aspect of statistics end up championing axiomatic deduction and the statisticians who embrace the science side end up championing (or at least acknowledging) induction.

  20. William/Matt/Briggs [delete whichever is inapplicable]

    In your paper you give several quotes. Unfortunately, I could not find the Popper quotes. I don’t possess LScD and the others appear to come from a book by PA Schlipp (The Philosophy of Karl Popper) that I also do not possess. I’m prepared to believe that they are correctly attributed to Popper. However, the quotes occur in the absence of context. I will supply some context from my knowledge of Popper’s works. First, to David Hume:

    “(a) We have no reason to believe any proposition about the unobserved even after experience!”

    This is Hume’s argument to support that statement(paraphrased):

    1. Currently observed matters of fact are known directly.

    2. Currently unobserved matters of fact are not known directly.

    3. Propositions about currently observed matters of fact do not logically imply propositions about currently unobserved matters of fact (though we usually assume that they do). We usually invoke the law of cause-and-effect, which is itself neither known directly nor logically deducible from anything known directly.

    4. Nothing else directly known implies any propositions about currently unobserved matters of fact.

    Therefore, we don’t know (we have no reason to believe)
    anything about currently unobserved matters of fact.

    “(b) There are no such things as good positive reasons to believe any scientific theory.”

    One virtually universal presupposition is that knowledge is true belief, but not mere true belief. For example, lucky guesses, or true beliefs resulting from wishful thinking, are not knowledge. Since on Popper’s account (and Bacon’s for that matter) scientific theories are plausible/useful guesses, they cannot be counted as knowledge. This is not to say they are not useful.

    (c) “The truth of any scientific theory is exactly as improbable, both a priori and in relation to any possible evidence, as the truth of a self-contradictory proposition” (i.e. It is impossible.)

    There are ever so many possible theories to explain any given set of phenomena. Since we cannot know with certainty that just this theory A is the one and only correct theory, then we can only state that the probability of its being the one and only correct theory is unity divided by the total possible number of competing theories. That is, it is improbable.

    (d) “Belief, of course, is never rational: it is rational to suspend belief.”

    Currently, there are two competing theories in physics, Quantum Mechanics and Relativity. Since they contradict each other in regard to gravity, then I can:

    1. Believe in Relativity, or
    2. Believe in Quantum Mechanics, or
    3. Suspend my belief and accept that while both theories are useful tools at least one is fallible.

    Briggs, it seems to me that you continually argue for less certainty about propositions. You seem to have reversed tack on this occasion. Perhaps I’m missing something.

  21. Joe,

    Once again, none of the criticisms about falsifiability we level are ours originally; they are shared widely in epistemology (our minor contribution is to apply these criticisms to probability models). Also, it is not our definition of falsifiable, it is the definition. To say something is false is to say that you have proof that it is not true. If you say something is “practically false”, you say that there is still some probability, no matter how small, that it is true, which means you cannot say it is false.

    Which leads into …

    PG,

    You do not have to believe in relativity of QM. You can say that either is probable, or that both give answers that are approximately correct. But this is all very loose language, which we have to tighten considerably if we want to put a probability measure on our statements.

    You’re right that I often argue for less certainty, but I do so for propositions about complex situations. For example, in politics or climatology, people often say incorrectly, “P is true”, where P is some proposition. They should actually say, “Given certain evidence P is probably true.” Or, really, in more complex situations, that “Given certain evidence, P might be true.” Which of course does not mean P is true.

    We’re also dangerously close to trying to have it both ways. If A = “something is true until we can prove it false” does that mean that A is true until we can prove it false? If not, then you agree with me that there are some things that are true.

  22. All,

    For all Popper proper-uppers, I proffer this quote from noted American philosopher John Searle:

    Those practicing scientists who took an interest in the philosophy of science at all tended to admire Popper’s views, but much of their admiration was based on a misunderstanding. What I think they admired in Popper was the idea that science proceeds by acts of originality and imagination. The scientist has to form a hypothesis on the basis own imagination and guesswork. There is no “scientific method” for arriving at hypotheses. The procedure of the scientist is then to test the hypothesis by performing experiments and reject those hypotheses that have been refuted.

    Most scientists do not, I think, realize how anti-scientific Popper’s views actually are. On Popper’s conception of science and of the activity of scientists, science is not an accumulation of truths about nature, and the scientist does not arrive at truths about nature, rather all that we have in the sciences are a series of do far unrefuted hypotheses. but the idea that the scientist aims at truth, and that in vsarious sciences we actually have an accumulation of truths, which I think is the presupposition of most actual scientific research, is not something that is consistent with Popper’s conception. [p. 23]

    This is from the essay “Philosophy in a new century”, which leads with the sentence, “The central intellectual fact of the present era is that knowledge grows.” [p. 4]. And “I believe the era of skeptical epistemology is now over. Because of the sheer growth of certain, objective, and universal knowledge, the possibility of knnowledge is no longer a central question in philosophy.” [p. 5]

    From John Searle, Philosophy in a New Century: Selected Essays, 2008. Cambridge University Press, London.

  23. To say, “A theory is true until proven otherwise” is nonsensical. If the theory is true, it is true. End of story.

    Mr Briggs, you missed my point. I was making a joke that tried to postulate a “relativist” truth. Unless you can prove that you have a direct communication with the “truth”, or with the very fabric of the universe, you simply can’t say that any theory is “true” as in “infallible”.

    Even Descartes knew this. He could only have one certainty.

    A direct consequence of this is that every theory we have is always relying on axioms demonstrated in another theory (Godel), and the only way we can have some sure that we aren’t that far away from the “truth”, is by measuring reality (empiricism).

    Otherwise, we would be swimming in a philosophical pool of Platonists, which we aren’t (or shouldn’t!).

    In this reckoning, I don’t take any theory to be “true”, and I also think that anyone that postulates these kinds of things is still a bit “naive” and afraid of the darkness of chaos and unknowns :).

    If an axiom is true based on no evidence—except by faith or inuition or similar word—then they cannot be proven false. So we have here examples of non-falisifiable statements that are just true. No empirical observation or additional (noncontradictory) premise can prove an axiom false.

    OK, so my last sentence was correct. I said,

    Or, you meant “True” as in “We take these as Given truths, as Self-evident Truths”. Well, if you take them as granted, then surely you aren’t interested in falsify them, now are you!!

    So, they are only falsifiable if you are really interested in investigating them. If you take them as faith, which I can define as “suspense of critical thinking”, then of course, they are unfalsifiable! That’s a tautology.

    From your last comment, I understand better where are you coming from. I’m holding my breath on this take. You aren’t convincing me. At least just yet. Einstein did prove Newton wrong, and Newton’s theory of Gravity was considered a scientific “truth”. As I previously and snarkingly said: “it’s true until proven otherwise”.

  24. PS: Because come on, mr Briggs, that’s the very interesting part of Science! The fact that Science has a self-correcting mechanism, which isn’t perfect, but it works with time, is essentially what makes it different than any theology or astrology self-contained “truths” and cargo-cult sciences! If you postulate that there exist “truths” that aren’t falsifiable, ever, then you’re not making science.

    In science, there can be no dogmas, except this one. And the more iconoclast the scientist is, while doing a better job at explaining and predicting the world, the more reverred he is.

    That is plainly obvious. Who are the most interesting scientists of the last centuries?

    Darwin, revolutionized our concept of Life.
    Einstein, revolutionized time, space and gravity.
    Bohr, revolutionized the scale of the very small (and outraged Einstein)
    Feynmann, who re-revolutionized the scale of the very small (and outraged Bohr).

    So, when you quote this:

    …but the idea that the scientist aims at truth, and that in vsarious[sic] sciences we actually have an accumulation of truths, which I think is the presupposition of most actual scientific research…

    …it’s just a simplistic view of what science is. Which works at most of the times. I mean, it’s very rare such revolutions as QMs, so most people can just work on top of those things.

  25. It seems Popper uses “falsifiable as a logical-technical term” as being falsifiable in principal, as a demarcation criterion of falsifiability.

    He does not use it to describe a theory which has been “demonstrably” falsified. He prefers the term “falsification.”

    He asserts the first refers to the “logical possibiliy of a falsification in principle, the second refers to a [italic] conclusive practical experimental proof (italic] of falsity. But anything like conclusive proof to settle an empirical question does not exist.

    Popper, Realism and the Aim of Science, ed. by W. W. Bartley, III, (London: Routledge, 1996, 1983) Introduction, 1982, p. xxii.

    Bartley reformulated the problem of induction, which Popper describes: “after having given a negative solution to the classical [italic] problem of justification. [italic], I have replaced it by the new [italic] problem of criticism, [italic] a problem for which I offer an affermative solution. Bartley says the transition is “fundamental.” (p. 20)

    Popper observed “Philosophers are not of course the only ones who believe in induction and in the existence of ‘inductive procedures’. As I said before, almost everybody does, including many psychologists—especially those interested in the theory of learning; many biologists; and quite a few physicists. As to the theory of learning, no doubt [italic] we can, and do, learn from experience.” [italic] (p. 39)

    These observations are disjointed, for which I apologise, but the seemed to bear on the question of whether he believed in induction regarding all problems. He didn’t as of 1982 when he wrote them. Certainly, he believed of the three ways of learning (1) trial and error, (2) by repetition or habit, and (3) by imitation or ‘asorbing a tradition, “only the first was relevant to the growth of knowledge.” (p.40)

    Thus, Popper did not deny the role of induction in other fields. Regarding the critical assertions in your paper of his role vis-a-vis statistics do you think, in light of the above, they still apply?

    Willi

    Willi

  26. It certainly may be the case that the definition of “falsifiable” presented in the paper is the definition and others are based on misunderstandings. If that’s the case, then I think you should include a reference to the given definition and the words of the people at whom you are addressing this argument that they use it that way.

    I only say this because the definition you present seems so transparently weak and that someone who wished to dispute your argument could simply say that he doesn’t agree with the definition.

    In any case, I don’t use the term falsifiable in the way you present. I get the impression from the comments that I’m not alone in that.

  27. This reminds me of the most famous solved math problem Fermat’s Theorem. Click here to see other unsolved problems. Are these unsolved problems true? I only know that there are mathematicians who are trying diligently to either come up with falsifying cases or prove them by deduction. In the end, mathematics is based on the proof by deduction.
    ______________

    Premise: If you are a statistician, then you are humorous. Brigg is humorous.
    Conclusion: Briggs is a statistician.

    The above argument is not invalid deductively. However, it may be understood as an inductive argument that makes a probable conclusion. As we have more evidence, a more probable conclusion could be reached. That is, an inductive inference has a probability property. I think it’s incorrect—I did not read your paper carefully and could very well have misunderstood you—if you are implying that Bayesian inference is valid deductively in the paper; see (9). It’s worth noting that deductive logic and mathematics are important for making good inductive generalizations.

    I have always told students mathematics is not a spectator sport. Now, hmmm…the philosophy of science could be a spectator sport (I am in no way saying that it’s not interesting).

    My worldview based on my enculturation! ^_^

  28. Probability is a measure of degrees of belief; isn’t it also a measure of degrees of falsifiablilty? Oh… I didn’t will this question out, it just occurred to me.

    Well, that’s it for now. It’s too nice outside to stay in front of the computer.

  29. All,

    No comments about Searle’s words?

    JH,

    Um. Your argument is invalid. The conclusion “Briggs is a statistician” is inductively stated and is not a deduction, as you say. If you had said in your second premise that “Briggs is a statistician” then you could have deductively said, “Briggs is humorous.” Inductive arguments are logically invalid, but that does not imply that they are always unreasonable (like the ‘flames’ example).

    Nope, I would never had said that the conclusions of probability arguments are in general proved deductively: I say quite the opposite. Most are stated inductively—including, I claim, most statements, or practical decisions, made by frequentists.

    Joe,

    The definition of “falsifiable” is weak? Hmm. Well, it is what it is. Do you have an alternate definition? I had thought that word was the least controversial aspect of the whole debate.

    Let’s hear from old man Popper himself (The Logic of Science, 1959, Routledge; all quotes from pp. 18-19; all emphasis original).

    “Now in my view there is no such thing as induction.” (Thus negating the ‘flames’ example: he would also not be able to say why you would not choose to jump off a 20 story building.)

    “Theories are, therefore, never empirically verifiable.” (Under which, I swear to God, he writes the following:)

    These considerations suggest that not the verifiability but the falsifiability of a system is to be taken as a criterion of demarcation. In other words: I shall not require of a scientific system that it shall be capable of being singled out, once and for all, in a positive sense; but I shall require that its logical forms shall be such that it can be singled out, by means of empirical tests, in a negative sense: it must be possible for an empirical scientific systems to be refuted by experience.

    He also, in a footnote to that statement and in a section further on in the book, strongly insists that falsifiability is not a criterion of meaning.

    On p. 19, he then admits the following (yes, this is really there): “[I]t is still impossible, for various reasons, that any theoretical system should ever be conclusively falsified. For it is always possible to find some way of evading falsification…” (he then gives examples how; but this at least makes his second statement, that “theories are never empirically verifiable” consistent).

    It was statements like these that lead Searle to say (p. 77; same book as below), “[M]any scientists…profess to admire Popper’s philosophy of science. I think they do not properly understand Popper…they do not understand that Popper rejects the basic assumption behind their enterprise; namely the assumption thhat they are getting at the truth.”

    I’ve also, in deference to Mr Git, been avoiding quoting Stove. But if I did…oh boy, sparks fly!

  30. JH,

    I do not say probability is a degree of belief, though I am aware that subjective Bayesians prefer that definition. I claim, with Carnap, Keynes, Jaynes, Stove, and others, that probability is a matter of logic. The conclusion of any argument may be given a probability measure. This measure is not always a single number; it is often an interval. If the conclusion is false, then the probability is 0, so, yes, probability can be used to measure falseness.

    For a more complete description, see this amazing new book (Chapter 1).

  31. Of course, I was wrong. I meant to say that “The above argument is not valid deductively. I was in too much of a hurry.

    No, you do not say probalbity is a degree of belief in the paper.

  32. William

    “Don’t forget that there are a set of axioms that lead to a consistent and complete theory of arithmetic. ”

    This is a falsifiable statement and one that has already been falsified 78 years ago .
    The Godel incompleteness theorem says that :

    “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent theory that proves certain basic arithmetic truths , there is at least one arithmetical statement that is true , but not provable in the theory. ”

    At that stage despite empirical observation that nobody has succeeded in proving simultaneously P and non P in 1000 generation of mathematicians all using the same set of axioms , it could still not be excluded that it would happen one day and that the mathematics , or better said the set of axioms used to generate the mathematics , would appear inconsistent .
    Thanks God it has been proven a few years after the Godel theorem that the first order (Peano) arithmetics is consistent .

    As all or almost all our mathematics are generated from Peano arithmetics , we are now in a situation where we know that as long as we carefully stay strictly within this frame , we can produce theorems with no fear that the whole building will crash one day on our heads but following the first Godels theorem we also know that these mathematics are incomplete .

    There is a fundamental difference between , say , mathematics and physical theories .
    In mathematics and I am here talking about mathematics based on Peano arithmetics we have the formal certitude of consistency and an easy way to prove most truths even if we know that we will not prove ALL the truths by using the tools (axioms) of the mathematics .

    In physics there is no warranty of consistency – f.ex like it has been mentionned in this thread , quantum mechanics and general relativity seem inconsistent .
    In fact I would relativize (no pun intended !) this statement because a more accurate statement would be that they don’t see the space time in the same way – for GR space time IS the physics while for QM space time is the independent stage where physics takes place .
    That is not exactly inconsistent but not very far either .
    And obviously physics doesn’t deal with formal systems but with things that fly , flash and push what makes them available to experimentation .

    That’s why for me the falsification process is very different in physics and in mathematics and I like to think that there will be one day a meta-Gödel theorem saying that “Any mathematical theory describing reality is either inconsistent or incomplete .” 🙂

  33. Tom,

    Right! I should have said, “first order arithmetic” or “first order logic”. However, now that we know it is in fact consistent, and true, it is not falsifiable.

    I’m right along with you on physical theories, too. These are often only probably true or probably false given certain information. Also, theories that offer statements that are only probably true cannot be falsified about the statements they make, unless the theory makes a statement like “X cannot happen: it’s probability, given the theory, is 0”. If we then see X, the theory is false.

    The example I gave earlier about a “normal distribution theory” is a good example where no observation can falsify the theory.

  34. William

    Yes , mathematics based on Peano arithmetics are consistent so non falsifiable even if incomplete .
    And as that is about whole of our mathematics , the question of falsifiability of mathematics as a whole is not an issue . A conjecture in the frame of mathematics may still be falsified unless it belongs to those unprovable truths but the mathematics as a whole are safe .

    As for the quasi probabilist physical theories (f.ex QM) , I am not sure that I follow you fully .
    If the QM says that in an experiment I would observe the spin projected on some axis equal to X with a probability of Cos (phi) where both the experiment and phi are perfectly defined , then I need only to run some 1000 experiments for different phis and measure the spin .
    If the spins measures which gave X don’t follow this law (especially for phi 0° where every measure must give X and 90° where none may give X) then the theory is falsified .
    Even if there is a single number like “The probability to observe the spin up is 50%” then when I observe 1 000 000 spins then about the half with the standard error
    99,9 % bars must show a spin up .
    If not then the theory is again falsified .

    But I think that I see what you mean , I can always have the bad luck to fall on the
    10^-47 probability where it is rather far from the half .
    Then as I am convinced by the soundness of the theoretical argument I will redo the experiment .
    If I have again bad luck and fall again on the 10^-47 probability I begin to have doubts and redo the experiment a third time .

    So yes , mathematically speaking , even 10^- much , is still not 0 but every physicist will admit that a 0,000001 % falsification is a 0 % falsification especially if the theoretical background is sound .
    That’s the problem with physics that there are never exact results like in mathematics so one has to substitute to the rigorous word “falsification” the words of “confidence in falsification” .

  35. But TomVonk, the thoery you describe contains “certitudes” (by the mapping of cos) mixed in with “probabilities.”

    A theory that is careful never to give any certitudes would not be formally falsifiable, which seems to be Briggs’ point; whereas our rational belief would clearly been strained at 99+% falsification.

    However, it does raise an interesting question whether most “probabilistic” theories do contain a some claims which are made (or assumed) with a probability of 1 or 0. It seems that they would, if you now compare the use of mathematical “axioms” with physical “axioms.”

Leave a Comment

Your email address will not be published. Required fields are marked *