March 11th, 2015
By Daniel Brockert
The Space of Belief Possibility- A response to the article
Why People “Fly from Facts.”
http://www.scientificamerican.com/article/why-people-fly-from-facts/
It has happened a million times. You’re debating a contentious issue at the dinner table, coffee shop or most likely on Facebook when someone tries to cool the tension by playing the ‘everyone has their own truth’ card. Aside from the fact that it is annoying, why don’t people pull the ‘everyone has their own truth’ shtick more often than they do? If you really believe that everyone has their own truth, why not argue with your math teacher for a better grade, since 2+2=5 for me. Taking the idea one step further, why even try to convince anyone of anything? Could it be that when people say that everyone has their own truth, they mean that the answers are objectively uncertain (that’s why people disagree) but the answers matter a lot?
Troy Campbell and Justin Friesen may have shed light on this behavior with new research outlining how people justify their beliefs in different situations.
In Scientific American Campbell and Friesen write:
Our new research, recently published in the Journal of Personality and Social Psychology, examined a slippery way by which people get away from facts that contradict their beliefs. Of course, sometimes people just dispute the validity of specific facts. But we find that people sometimes go one step further and, as in the opening example, they reframe an issue in untestable ways. This makes potential important facts and science ultimately irrelevant to the issue.
It’s worth taking a moment to reflect on two aspects of our beliefs and how they differ by domain. In the case of basic mathematics, I am certain than 2+2=4. But I feel no particular passion or conviction about it and the belief that 2+2=4 plays no role whatsoever in shaping my identity. Why is that so? Wouldn’t I want certain foundations on which to rest my convictions and to identify myself as a person who believes in arithmetic? But people never have strong convictions about the truth of arithmetic or count themselves as proponents of “arithmetic fundamentalism” even though this is obviously the case. Admit it - if you’re really so open-minded then why won’t you at least entertain the possibility that 1+1= 1,003,533?
Michael Minnich and I have a proposal: Beliefs have to differ among individuals or groups in order for people to feel conviction or in order for beliefs to shape people’s identities. Identity necessarily entails that some people are included as members in a group and other people are excluded. As a result, brute facts can play no useful role in identity formation. People are more likely to feel conviction when their beliefs are challenged by competing beliefs, rejected or otherwise threatened. A growing body of experimental evidence supports these claims.
Why are politics and religion so important in our convictions and in the shaping of our identity? It’s a trivial observation that people’s religious and political views differ. One of the major uncertainties of life is grasping with the fact that on some political, moral and spiritual questions there are no certain answers. We may feel certain but we always maintain an awareness that there are others who disagree with us and this disagreement is threatening. Being around like-minded people (politically or spiritually like-minded, but not mathematically like-minded) is a liberating experience for those of us who feel strong convictions about objectively-uncertain matters. Trying to convince others to believe like us in uncertain domains is partly a test - do you belong in our group or are you an outsider? When the anti-vaxxer fails to convince me that vaccines are dangerous, it is clear that I’m an outsider that must be excluded. Unfriend.
Beliefs act as a group membership litmus test. If you believe as I do already, “great! My brother!” If I can change your mind, “welcome to the club!” However, if you are closed-minded and inconvincible, “get out!” (Note how closed-mindedness is almost always a trait someone else has).
We assert that there are objective constraints on our subjective experience of belief possibility and that given a set of data there may be no alternative possible belief. As William James observed in his article The Will to Believe:
Does it not seem preposterous on the very face of it to talk of our opinions being modifiable at
will? Can our will either help or hinder our intellect in its perceptions of truth? Can we, just
by willing it, believe that Abraham Lincoln’s existence is a myth, and that portraits of him in
McClure’s Magazine are all of someone else?
Many aspects of life, including controversial issues in the sciences, don’t fit the above pattern. There is often more than one possible belief given a set of data. Despite the room for disagreement, one idea often dominates. The philosopher of science Paul Feyerabend criticized the way an idea in the sciences can dominate merely because it came first, despite the fact that alternative interpretations could do an equally good job of explaining the known facts. Given the many instances in history in which well-established theories were overturned in scientific revolutions, he argued against what he called the consistency condition and in favor of theoretical pluralism. As he observes in his book Against Method:
(The Consistency Condition) eliminates a theory or a hypothesis not because it disagrees with
the facts, it eliminates its because it disagrees with another theory, moreover, whose confirming
instances it shares… “The first theory has the right of priority over equally adequate aftercomers.” It contributes to the old and familiar not because of any inherent advantage in it, but because it is old and familiar.
Feyerabend’s idea was a critique of scientific dogma, but it could just as easily apply to any aspect of life in which there is more than one reasonable belief that could be held given a set of information. Taking the belief possibility principle one step further - what about aspects of life that are both untestable and open to multiple interpretations? In cases like these, there are a potentially infinite number of beliefs that could be held given the available data. We assert that in the cases of political ideology and spirituality, this is exactly what happens. Furthermore a rough mathematical model can be created to represent these different levels of theoretical uncertainty.
In cases of brute facts, there is no objective uncertainty and thus there is no space for alternative belief. In cases of scientific uncertainty, there is space for alternate belief that varies with the problem set presented. For hard sciences such as physics, there is often a very limited space for possible belief. We know from experimental observation that objects fall at a fixed rate. Due to repeated testing we are fairly certain that our results are accurate, thus there is a very small belief possibility space (few opportunities for alternative belief). Belief possibility spaces aren’t always limited in physics, however. Interpretations of the double-split experiment are quite varied (in a recent poll of physicists, the most popular interpretation, the Copenhagen interpretation, only garnered the support of 43% of physicists with the remainder of the vote split among competing hypotheses). http://www.nature.com/news/experts-still-split-about-what-quantum-theory-means-1.12198
Compared with other sciences, however, physics is an easy case. Sociology, economics and linguistics, sciences that deal with complex data sets and multiple interacting variables, are constantly battling to distinguish between mere correlation and causation. Because of the higher complexity of causal variables dealt with in the social sciences, they occupy a high level in the belief possibility space.
I have represented the concept in a diagram here:
The BPS is divided into 3 main sections, though one could further subdivide these levels if necessary. Brute facts are obvious truths that allow no space for alternative belief. In the section of observable reality, hard sciences have room for skepticism but the space for alternative belief is cramped compared with social sciences where causal variables are far more difficult to isolate. (Medicine may be seen as on par with the social sciences in this regard) Political ideologies, which deal with observable reality, but often rest on untestable claims, straddle the line between observable and unobservable reality. Spiritual claims, because they are untestable, occupy the unobservable reality section of the Belief Possibility Space. Beliefs at this final level must be taken on faith.
Belief Possibility Spaces, like Bayes’ theorem, are an attempt to model beliefs mathematically. New scientific knowledge, according to our hypothesis, is synonymous with closure of the Belief Possibility Space for a given claim. Prior to the invention of the telescope, it was an open question whether the geocentric or the heliocentric theory was correct. There were multiple plausible explanations given the available data. The invention of the telescope, by providing additional data, falsified the geocentric theory and closed the belief possibility space for that domain. The heliocentric theory is now an unchallengeable scientific fact. The more room for doubt or alternative belief, the larger the belief possibility space.
When there is no reasonable alternative hypothesis given a set of data, as is the case for brute facts, the Belief Possibility Space value is 0. When there are an infinite number of possible hypotheses given a set of data (as with purely religious/spiritual matters) the BPS value is 1. Science necessarily deals primarily with the dimensions of the BPS that have values somewhere between 0 and 1.
Does reality exist? ← This question has a BPS value of 0
Why does reality exist? ← This question has a BPS value of 1
Belief Possibility Space equation:
1 - 1/n' = Dimension of BPS
1 - 1/n' = Dimension of BPS
1= Infinite Uncertainty
0= Uncertainty level of zero
H = Set of reasonable hypotheses given a set of data. The bigger a set H, the higher in
BPS an individual hypothesis h will be.
H= {h₁, h₂, h₃,h₄…..}
h= Individual hypothesis being considered
k = Lack of a positive claim (Rational/Scientific skepticism)
R= Set of Radical Skeptic Objections (i.e. Does the external world even exist? Maybe the pyramids were built by aliens rather than people, maybe a ghost committed the murder while the husband was in the other room and stole some of the husband’s DNA and left it on the body) Because of the implausibility of radical skepticism, it receives a value of zero in the denominator of the BPS equation.
H' = H + k
n' = all possible hypotheses
n' = H' + R
n' = all possible hypotheses
n' = H' + R
One objection to the model is the definition of hypothesis. Because hypothesis categories have fuzzy boundaries and because a belief could be subdivided in a potentially infinite number of ways, this can lead to infinite regress. 2+2=4, but what is 2 anyway? How do I know that I have the same notion of 2 that you do? This type of objection could arise in a court case that involves a run stop sign. What does “road” mean? In order for the law to cover a run stop sign it must have happened on a road. Was it a road? However, if one sticks to a common-sense interpretation of belief, we believe a BPS model can provide an advantage over Bayesian models. We prefer the use of the term “data” rather than “evidence” to avoid loaded assumptions about what does and does not count as evidence.
Comparison with Bayes’ Theorem
Bayes’ theorem provides a model for what we are supposed to believe given new evidence. In Bayes’ theorem new evidence and new probability judgements are weighed against old evidence and old probability judgements. Bayes’ theorem asks “what should we believe given some new evidence?” This leaves open the question, “why did we believe what we believed before?” Presumably what we believed before was a belief we had in light of evidence that was new to us before. This leads to an infinite regress problem. Our Post-Bayesian Belief Possibility Space approach asks the question in a slightly different way: “What could we believe now given the total evidence we have now? What could we have believed before given the total evidence we had before?” By casting aside the issues with prior and posterior probabilities, we hope to provide a notion of conditional probability that overcomes the philosophical problems in the Bayesian paradigm while greatly expanding the scope of beliefs the paradigm covers. We hope philosophers and mathematicians can develop this notion further.
Let us finish with a final comment about the study by Troy Campbell and Justin Friesen. In Scientific American they write:
While it is difficult to objectively test that idea, we can experimentally assess a fundamental question: When people are made to see their important beliefs as relatively less rather than more testable, does it increase polarization and commitment to desired beliefs? Two experiments we conducted suggest so.
In the experiment they manipulated people’s intuition about how testable certain beliefs were. In doing so they were playing with people’s intuitions about uncertainty and the space for possible belief. One possible avenue to follow is testing the boundaries of the experiment tactic itself. What intuitions do we have about certainty and uncertainty that can’t be easily manipulated? Can people be led to believe that basic arithmetic is a controversial point of view? Or as Woody Allen once joked “I dated a philosopher girlfriend. She convinced me that I don’t exist.” What aspects of our beliefs fit William James’ observation that they simply could not be modified at will?
In conclusion, objective uncertainty constrains the possible beliefs we could hold in a given domain. In basic arithmetic, objective uncertainty is zero, whereas in spiritual matters there is infinite uncertainty (requiring a leap of faith for the religious). These objective differences in certainty affect us in a counter-intuitive way. We feel no conviction on objectively certain matters and these matters play no role in shaping our identity. The matters that shape social identity the most are objectively uncertain matters precisely because they are uncertain. The more possible ways a set of data can be interpreted, the more opportunities for social difference the data provide. Now for an experiment in brute fact belief. Can you make yourself believe this article is not finished?
Comments (and brutal criticism) are welcome!
Email Daniel Brockert at
daniel.brockert@yahoo.com
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