Gradualism is the cornerstone of Darwin’s Theory of Evolution because without it, he could not justify one organism evolving into another. ‘Gradualism’ equals ‘Continuity’ and nothing more. It does not describe the direction, the speed or anything else we would like to know about that process.
In math, a function is gradual if continuous. A continuous function has a Grade’ (Slope) at every point. If a function is not gradual (continuous), then it is Discrete and has no ‘Grade’ (Slope). A Discontinuous function is a special case of ‘Continuous over limited ranges’.
Is Nature Gradual? No, Nature is Discrete from the most elementary particles, to molecules, cells, and organisms. New organisms are created by discrete processes and result in newborns that are measurably different from each parent while all DNA mutations are discrete events. Gregor Mendel observed the discrete nature of biology as early as 1865 in the inheritance of dominant and recessive alleles. Darwin might have learned that from Mendel’s papers sent to him, had he read and correctly interpreted the results.
We classify organisms into distinct groups with little if any overlap and with significant homogeneity within the group. If Gradualism were the norm, all living animals would fill a continuous spectrum which would make their classification in various taxa completely arbitrary.
Biological Time Continuity (Gradualism or Common Descent) would require a similar belief in Biological Space Continuity (for present time) were it not for this being contrary to observation. Instead of Gradualism, we observe that even unicellular organisms with huge populations and short-lived generations do not occupy a biological continuum. Plant diversity over the altitude & latitude continuum is a good example of Discontinuity in Nature: as conditions change, we see a changing mix of distinct species, rather than hybrid species as would be expected if Gradualism were true. Animal territoriality is also an example of discrete successful designs dominating certain ranges and mixing with each other at range boundaries without significantly changing their characteristics.
What about Speciation and Hybridization? A certain flexibility appears built into each biological design – more in some than in others. What we call Speciation and Hybridization may in fact be no more than adaptations within these flexibility ranges.
Doesn’t the Fossil Record show Gradualism? Without confirming experiments on living organisms, it is impossible to determine whether the Fossil Record shows Gradualism or instead predisposition to Gradualism prompts an incorrect interpretation of the Fossil Record.