## AbstractModern science are said to be anti-platonic but yet Fisherian theoretical distributions are pervasively used in statistical inferences. The purpose of this article is to determine the existence of theoretical distributions subsumed within mathematical reality, and highlight several meaningful implications for psychological researchers to consider in the application of statistical procedures. Therefore, we begin with a brief overview of the misinterpretation of statistical testing, followed by a discussion of mathematical reality. In addition, ideas for and against theories of mathematical reality by noted scholars are presented, and inconsistencies within these theories are elucidated. Finally, the conclusion highlights practical implications for researchers when implementing statistical procedures.
## IntroductionIn statistics, we apply theoretical distributions to determine the significance of a test statistic. However, Sir R. A. Fisher, the founder of statistical hypothesis testing, asserted that theoretical distributions against which observed effects are tested have no objective reality "being exclusively products of the statistician's imagination through the hypothesis which he has decided to test." (Fisher, 1956, p.81). In other words, he did not view distributions as outcomes of empirical replications that might actually be conducted. In a similar fashion, Lord (1980) "delinked" the statistical world and the real world:
## Theories of Mathematical RealityDiscussion of mathematical reality pertaining to statistics is very rare. Most mathematicians and philosophers center discussions on geometry, algebra, and other branches of mathematics. Therefore, philosophy of mathematics seems to be remote and even irrelevant to social science researchers. Although the following review of major theories of mathematical reality is not directly based on statistics, their implications are still important to the applications of theoretical distributions.
## Platonic WorldsMost philosophers relate mathematical realism to Platonism
## Wittgenstein's Approach
## Russell's ApproachRussell (1919) disagreed with intuitionistic approach and affirmed the existence of unchanged structures in mathematics. In philosophy of science, his leading motive is to establish certainty in an attempt to replace the Christian faith he rejected. Russell found certainty in mathematics, because he believed that mathematical objects are eternal and timeless (Hersh, 1997). In Russell's view, in order to uncover the underlying structures of these eternal objects, mathematics should be reduced to a more basic element, namely, logic. Thus, his approach is termed
## Godel's TheoremKurt Godel, the great mathematician who was strongly influenced by
## Mandelbrot Set
## God's RevelationOxford mathematician Penrose (1989) looked at the
## The World is RoundThe preceding argument made by Penrose resembles what Socrates raised thousands of years ago. Socrates endorsed the idea that knowledge is grounded on generally agreed definitions (so-called "What is F?" question) and a common frame of reference rather than an individual's perception. If knowledge is just perception, then no one can be wiser than any one; one can be his own judge and any dialog between people is impossible (Copleston, 1984). In defending mathematical realism, Drozdek and Keagy (1994) made a similar assertion:Realism keeps mathematicians on guard much more than the intuitionist approach would do. For a realist, there is this objective sphere which is an ultimate yardstick of a theory's validity. For an intuitionist, the intuition is the ultimate guide, and if distorted, or turned from well established logical principles, it has nothing, even in theory, to found this statement. (p.340)
## DiscussionAccording to Plato, Russell, Mandelbrot, and Penrose, mathematical reality does exist, which implies the existence of theoretical distributions. However, equally learned scholars such as Wittgenstein and Godel do not believe the existence of mathematical reality can be proved. Which scholar should be believed? Each theory is important to the debate of the existence of mathematical reality and theoretical distributions; each theory embodies valid and invalid assertions. Beginning with Russell and Whitehead, the discussion will focus on the inconsistencies within each theory. Following the discussion, the conclusion will highlight several practical implications for researchers to consider when implementing statistical procedures.
## ConclusionWe cannot answer whether theoretical distributions exist or not. However, statistical procedures are being applied in psychological research every day, and therefore a firm understanding of their rationale and appropriate application are of the utmost importance. There are three implications to social scientists pertaining to the discussion of theoretical distributions:
## Notes
## ReferencesBlack, M. (1959).
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