Amaryllis is a form of lily and as such its petals are in sets of three. It has two such sets, one fore and one aft. One set may be roughly identified by direction as north, southeast and southwest. The other set as south, northeast and northwest. These sets are natural measureable properties of the amaryllis plant and are useful in the science of taxonomy.
The mathematics of sets may be characteristically embodied in nature and when so, it forms part of the base of science. However, that is not to say that the entirety of the mathematics of sets is characteristically embodied in nature. One important exception is the mathematics of probability.
Probability is the fractional concentration of an element in a logical set. The mathematics of probability is centered in the formation of other logical sets from a source set, where all sets are identified by their probabilities, i.e. their fractional compositions.
In discussing the mathematics of probability where the source set is one of two subsets of three unique elements, it would not be inappropriate to refer to the amaryllis flower as a visual aid in discussing what is essentially logical and in no way characteristic of the nature of the amaryllis flower.
In randomly forming sets of two petals where the source set is the amaryllis flower, what are the probabilities of the population of sets defined in terms of fore and aft petals? The population defined is of four sets of two petals each. One set consists of two fore petals. One set consists of two aft petals. Each of the other two sets of the population consist of one fore petal and one aft petal. The probability of sets of two fore or two aft petals in the population of sets is 25%, while the probability of a set of one fore and one aft petal is 50%.
One could imagine placing one set of six petals from an amaryllis flower in each of four hundred pairs of hats and blindly selecting a petal from each of two paired hats. One would expect the distribution of paired sets of petals to be roughly one hundred of two fore petals, one hundred of two aft petals and two hundred of one fore and one aft petal. This would represent a material simulation of the purely logical concept of probability.
What is the probability of a set of eighteen randomly selected amaryllis petals containing at least one ‘north’ petal? The answer, P, equals (1- ((n – 1)/n)^x), where n = 6 and x = 18. P = 66.5%.
Notice that these questions in the logic of probability have nothing to do with amaryllis flowers or petals or their manipulation. The petals and their visual (or actual) manipulation are entirely visual aids in a discussion of pure logic. It is materially impossible to select any material thing at random from a set of material things. Selection is always explicable in terms of the material forces involved. Thus, it is always non-random. It is by convention that we equate human ignorance of the details of the material process of selection with mathematical randomness.
We say that the probability of one sperm fertilizing a mammalian egg is one in millions. What we mean is that the fractional concentration of any one sperm is one in millions and that we are ignorant of the detailed non-random physical, chemical and biological processes by which one sperm of the natural set fertilizes the egg.
It is, of course, permissible to use the mathematics of probability in many instances when for any number of reasons we are ignorant of the scientific explanation of material processes. However, we must be constantly aware that the mathematics of probability characterizes human ignorance and not material reality when it is used as a tool to compensate for a lack of knowledge.
The mathematics of probability is an exercise in logic, unrelated to the nature of material things and their measurable properties. In contrast, science is the determination of the mathematical relationships among the measureable properties of things, which properties are characteristic of the nature of material things.