Table 2.
Most common quantitative-based research philosophies: assumptions and stances and the likelihood of embracing the quantitizing process.
| Research philosophy | Ontology | Epistemology | Axiology | Methodology | Likelihood of quantitizing | Rationale |
|---|---|---|---|---|---|---|
| Platonism (Mathematical Realism; Balaguer, 1998; Tomšič, 2017) | Abstract mathematical entities exist independently of human thought; timeless, non-physical objects. | Knowledge is attained through intellectual intuition and logical reasoning. | Values the discovery of absolute and universal truths. | Deductive reasoning and discovery of eternal truths. | High | Quantitizing complements the search for universal mathematical truths. |
| Nominalism (Mathematical Constructivism; Kerkhove and Van Bendegem, 2012; Szabo, 2003) | Denies the independent existence of mathematical entities; sees them as constructs or labels. | Knowledge is a human construct, dependent on social practices and language. | Values practical utility and coherence. | Constructive mathematics and verification through consistency within mathematical systems. | Low | Prioritizes human-centric constructs over objective quantification. |
| Structuralism (Sturrock, 2008) | Emphasizes the structures or relationships among mathematical entities, rather than the entities themselves. | Knowledge arises from understanding these structures. | Values insight into the structural aspects of mathematics. | Analysis of relationships and patterns within mathematical systems. | Moderate | Useful for illuminating structural relationships. |
| Intuitionism (Dummett, 2000) | Mathematics is a mental construct, not reflecting any external reality. | Knowledge is subjective, accessed through mental processes and intuition. | Values the certainty and constructiveness of mathematical proofs. | Constructive proofs, emphasizing processes that can be intellectually grasped. | Low | Due to the focus on mental constructions rather than empirical data. |
| Formalism (Detlefsen, 2007) | Mathematics is about manipulating symbols according to agreed rules; the symbols don't necessarily represent real objects. | Knowledge is based on mastering these formal systems and operations. | Values logical consistency and rigor in formal systems. | Development and exploration of formal systems, independent of their interpretation. | High | Aligns well with the manipulation of formal systems and precise measurements. |
| Logicism (Demopoulos, 2013) | Mathematics can be reduced to logical foundations. | Mathematical truths are derived from logical truths. | Values the clarity and undeniable truth provided by logic. | Reducing mathematics to logic to prove mathematical truths. | High | Quantitizing supports the objective and universal nature of logical structures. |
| Empiricism (Meyers, 2006) | Mathematical knowledge is derived from experience and is empirical. | Knowledge is provisional and empirically tested. | Values empirical verification and practical applications. | Empirical observation and experimentation. | High | Crucial for linking mathematics to empirical observations. |
| Finitism (Ye, 2011) | Only finite mathematical constructs exist; rejects the existence of actual infinity. | Knowledge is about finite procedures and their results. | Values computability and concrete results. | Restricts mathematical practice to finite operations. | Moderate | If the focus remains on finite and tangible outcomes. |
| Realism (House, 1991) | Mathematical entities exist independently of human knowledge or perception. | Knowledge of these entities is discovered, not invented. | Values the discovery of objective, independent truths. | Objective investigation and logical analysis. | High | It aids in the objective analysis and understanding of mathematical entities. |
| Anti-Realism (Brock and Mares, 2006; Chalmers, 2009) | Denies the objective existence of mathematical entities outside of human conceptual schemes. | Mathematical truths are dependent on human practices or conceptual frameworks. | Values the practical and explanatory power of mathematics. | Focuses on the usefulness and practical application of mathematical concepts. | Low to moderate | Depending more on its practical utility than on seeking objective truths. |
| Fictionalism (Fine, 1993; Suárez, 2008) | Mathematical entities are akin to fictional characters; they do not exist. | Mathematical truths are “pretended” for their utility in explaining and predicting phenomena. | Values the usefulness and explanatory power of mathematical constructs. | Utilitarian use of mathematics as a tool for explanation and prediction. | Moderate | Appreciated for its practical benefits rather than its truth. |
| Psychologism (Crane, 2014) | Mathematics is a product of human thought and psychological processes. | Mathematical knowledge is derived from and limited by human cognitive capacities. | Values the understanding of human cognitive processes in mathematics. | Psychological investigation into how mathematical thoughts and processes develop. | Low | Focuses more on qualitative insights into human cognition. |
| Objectivism (Peikoff, 1993) | Reality exists independently of consciousness; specific principles govern reality, including mathematical ones. | Knowledge is based on objective observation and rational integration. | Rational inquiry and empirical evidence. | Rational inquiry and empirical evidence. | High | It aligns with the pursuit of objective knowledge through rational and empirical means. |
| Postpositivism (Phillips and Burbules, 2000; Popper, 2002) | Acknowledges that scientific knowledge is imperfect and theory-laden. | Knowledge is provisional and subject to revision; emphasizes critical testing of theories. | Values rigorous testing, critical thinking, and acknowledges the fallibility of scientific inquiry. | Scientific methods with a recognition of their limitations; uses qualitative and quantitative research. | High | But with a critical stance, recognizing the limits and potential biases of quantitative methods. |
Adapted from “Philosophical assumptions and stances of the most common mixed methods research-based research philosophies,” by Onwuegbuzie, 2024. Dialectical Publishing, p. 13. Copyright 2024 by Dialectical Publishing.