Starting with hopes, the principal meaning for ‘hope’ given in Webster’s New World Dictionary is: “a feeling that what is wanted will happen; desire accompanied by expectation.” Equivalently, therefore, ‘hope’ is what mathematicians call “expected value”, philosophers and economists call “utility”, engineers call “risk”, and gamblers call “payback”. Thereby, ‘hope’ can be estimated quantitatively: if some value (e.g., a monetary value) can be placed on what’s desired, then ‘hope’ is the assigned value multiplied by the probability that what’s desired can be achieved.

It might be useful to illustrate the method with a couple of examples.

For the first example, suppose you assign the value of “having a good day” to be $100. Further, suppose you estimate that the probability that you’ll have a good day, today, to be quite large, namely, 90%. Then a quantitative evaluation of your hope that today will be a good day is the expected value = (value) x (probability of its being achieved) = $100 x 90% = $90; i.e., you have quite a large hope that today will be a good day.

As another example, suppose that you assign the value of your life to be $1 billion, that you have cancer, and that the best information available suggests that there’s only 1 chance in a million that the therapy will be successful. Then the hope that you’ll assign to your recovery has a value of only $10^9 x 10^(-6) = $1,000.

Now, consider fears. Webster’s prime definition of ‘fear’ is

a feeling of anxiety and agitation caused by the presence or nearness of danger, evil, pain, etc; timidity; dread; terror; fright; apprehension…The Oxford-American Dictionary gives for ‘fear’:

A definition of ‘fear’ that I think improves those definitions and that reveals the symmetry (or anti-symmetry) of hope and fear is the following:

- an unpleasant emotion caused by the belief that someone or something is dangerous, likely to cause pain, or a threat…
- (archaic) a mixed feeling of dread and reverence
- (fear for) a feeling of anxiety concerning the outcome of something or the safety and well-being of someone
- the likelihood of something unwelcome happening…

fear is a feeling that what isn’t wanted (i.e., what's dreaded) will happen; dread accompanied by expectation.If that proposed definition of ‘fear’ is accepted and if values and probabilities can be estimated, then as in the case of hope, fear can be evaluated quantitatively.

To illustrate, consider examples similar to those used above.

For the first example, suppose you continue to assign the value of “having a good day” to be $100. Further, suppose you estimate that there’s a 10% probability that, today, you’ll have a bad day. Then a quantitative evaluation of your fear that today will be a bad day is the expected value = (value of a good day) x (probability of not having a good day) = $100 x 10% = $10; i.e., you’ll have only a relatively small fear that you’ll have a bad day today.

As a second example, consider again the case that you have life-threatening cancer. Suppose that you continue to assign the value of your life to be $1 billion and that there’s only 1 chance in a million that the therapy will be successful, i.e., that the probability of your dying is 0.999999 . Then the fear that you’ll die can be quantitatively evaluated as (value of life) x (probability of dying) = $10^9 x (0.999999) = $999,999,000, i.e., your fear of dying is very large.

If the above, proposed definition of ‘fear’ is accepted, then at least in simple cases and for a single event (or possession or similar), a link becomes apparent between fear and hope and with the assigned value of the event. Mathematically, the relationship can be seen starting from the following steps:

(i) Hope = Value x (probability of realizing that value), and

(ii) Fear = Value x (probability of failing to realize that value).

Now and for what follows, consider those relatively simple events (or possessions or similar) for which the value either materializes or doesn’t materialize, i.e., ignore “partial realizations”. In such cases, from the definition of ‘probability’:

(iii) Probability of realization = [ 1 – (probability of failure of realization)].

So, substituting the definition (iii) into either (i) or (ii), e.g., into (i):

(iv) Hope = Value x (probability of realizing that value)

= Value [ 1 – (probability of failing to realize that value)]

= Value – Value x (probability of failing to realize the value)

= Value – Fear, or

(v) Hope + Fear = Value .

In words, the result is that, for a rational person, the sum of the magnitudes of one’s ‘hope’ and ‘fear’ associated with a single event (or possession or similar) must be the total value placed on the event. Therefore, in that sense and for such cases, hope and fear are complementary. The result is a quantitative restatement of Spinoza's qualitative assessment:

There can be no hope without fear, and no fear without hope.As an illustration, consider again the hope that you’ll have a good day. Above, I estimated this hope to be (value = $100) x (probability of having a good day = 90%) = $90. Elsewhere above, I also evaluated the magnitude of the fear that you would have a bad day to be (value of good day = $100) x [probability of having a bad day = (1 – probability of having a good day) = 10%] = $10. Thus, consistent with result given by Equation (v), above, your hope of having a good day (= $90) plus the magnitude of your fear of not having a good day (=$10) sum to the value you placed on having a good day (i.e., $100).

More complicated illustrations deal with the fear of Hell, the hope of not going to Hell, the hope of Heaven, and the fear of not going to Heaven. As a first step, consider fear of Hell. For this case, suppose that the value I place on my not going to Hell was the huge sum $10^(100) and suppose I put the probability of my going to Hell (based on the likelihood of such a clerical concoction actually existing) to be 10^(-500). Then the magnitude of my fear of going to hell would be $10^(100) x 10^(-500) = $10^(-400) = 0.000000 (continue on for a total of about 400 zeros)…1 ¢; i.e., not worth a second thought. The other fears and hopes associated with Heaven and Hell are similarly not worth a second (or a second’s) thought.

Thus in practice, obviously the key steps in realistically evaluating hopes and fears are to define realistic values for what is desired or dreaded, respectively, and to obtain realistic estimates for the probabilities that what’s wanted (hoped) and what’s dreaded (feared) will be realized. In contrast, those people who chase the will-o’-the-wisp of Heaven or tremor at the thought of Hell are like children of the age of “the terrible twos”: adamant about knowing what they like and dislike but incapable of realistic evaluations of associated probabilities; they’re driven by emotion untempered by reason.

Undoubtedly, much of the blame belongs to the clerics of the world. Either they’re too ignorant to evaluate relevant probabilities or, similar to all con-artists, for their own profit they purposefully ignore relevant probabilities and promote resulting, unrealistic hopes and fears. In the end, though, adults who are taken-in by such snake-oil salesmen must accept responsibilities for their own lives: as all con-artists know, you can’t cheat an honest man. Yet, pity the poor children of the world (especially in Muslim countries): before they’re able to realistically evaluate probabilities on their own, they’re indoctrinated with an absurd hope of Heaven and a horrible fear of Hell.

Surely to any loving god (if only there were one!) a special place in Hell is reserved for clerics who concocted and promote the hideous idea of Hell. As Robert Ingersoll said:

All the meanness, all the revenge, all the selfishness, all the cruelty, all the hatred, all the infamy of which the heart of man is capable, grew, blossomed and bore fruit in this one word, Hell.Before 2000 BCE Ptah-Hotep admonished against it: “Let no man inspire men with fear…” Four thousand years later Ingersoll added:

Fear paralyzes the brain. Progress is born of courage. Fear believes – courage doubts. Fear falls upon the earth and prays – courage stands erect and thinks. Fear retreats – courage advances. Fear is barbarism – courage is civilization. Fear believes in witchcraft, in devils and in ghosts. Fear is religion – courage is science.As science continues to progress, I expect that (within a century or so) humanity will be able to purge itself of essentially all ideas about any supernatural Heaven and Hell. More quantitatively, since I place the value of purging humanity of such ludicrous ideas to be very large, say $10^(15), then even if the probability of it occurring within a century is only one in a million, then the value of my associated hope for humanity is $10^15 x 10^(-6) = $10^9 = one billion dollars – which, for me, reflects quite a bit of hope! And I admit to being an optimist, i.e., to focus on the hope that humanity will succeed in ridding itself of all organized religions rather than on the fear that it won’t.

www.zenofzero.net

All of this math makes me feel lucky that I'm happy and full of hope everyday! :)

ReplyDeleteI'm 'happy' to hear from you!

ReplyDeleteYours is a sparkling, crystal-clear blog. :)

In friendship, maybe I should add a word to the wise: if you do 'fear' math, you'd probably be happier if you stayed away from my book (at http://zenofzero.net ), especially the chapters on 'hope', 'probability', 'indoctrination', and 'truth'. :(