Continuous

In philosophy, “continuous” refers to an unbroken, uninterrupted whole without gaps, a concept applied in metaphysics and the philosophy of mind. It can describe the continuous nature of reality, as in the idea that the universe is a continuous spectrum of forms without discrete gaps, or it can define personal identity over time through a continuous stream of psychological states, such as memories and beliefs. This is often contrasted with the concept of “discrete,” meaning separate and distinct.
Core concepts

Principle of Continuity: The belief that reality is composed of an infinite series of forms that blend into one another, with a hierarchical order from the simplest to the most complex, as argued by ancient and modern philosophers like the Stoics.
Psychological Continuity: A theory of personal identity that posits a person’s identity is maintained over time by a continuous chain of psychological states, not by an unchanging essence.
Continuity vs. Discreteness: The ongoing debate about whether reality is fundamentally a continuum (continuous) or composed of discrete, separate units (discrete).
Continuants vs. Events: A distinction made in metaphysics between continuants, which persist through change, and events, which do not.

Applications in different areas

Metaphysics: The principle of continuity is central to discussions about the nature of reality, particularly regarding space, time, and matter. Philosophers like the Stoics argued that the cosmos was a continuous medium, a “plenum,” with no void, and that all motion and matter were linked through a continuous substance called pneuma.
Philosophy of Mind: The concept is used to understand the self and consciousness. Psychological continuity theory suggests that your identity persists over time because your present self is psychologically connected to your past self through a series of overlapping states, like memories and intentions.
Mathematics: While not a philosophical term itself, the philosophical and mathematical concept of continuity are deeply interconnected. In both fields, continuity implies the ability to be infinitely subdivided, unlike a discrete collection. This is seen in the historical debate over the existence of infinitesimals and the attempt to reduce continuity to discreteness.

AI responses may include mistakes.

[1] https://www.britannica.com/topic/priniciple-of-continuity

[2] https://fiveable.me/key-terms/intro-philosophy/psychological-continuity-theory

[3] https://plato.stanford.edu/entries/continuity/

[4] https://plato.stanford.edu/entries/continuity/notes.html

[5] https://digitalcollections.dordt.edu/cgi/viewcontent.cgi?article=2150&context=faculty_work

[6] https://plato.stanford.edu/archives/win2018/entries/continuity/

[7] https://www.rep.routledge.com/articles/thematic/continuants/v-1

[8] https://www.cambridge.org/core/books/concept-of-motion-in-ancient-greek-thought/aristotles-notion-of-continuity-the-structure-underlying-motion/418C1F1FFFB74DC77D8241636953B57D

[9] https://www.pdcnet.org/scholarpdf/show?id=saap_1998_0026_0081_0017_0018&pdfname=saap_1998_0026_0081_0017_0018.pdf&file_type=pdf

The concepts of the continuous and infinitesimal in mathematics and philosophy are deeply interconnected, with the continuous representing an unbroken whole and the infinitesimal representing quantities that are infinitely small. Historically, the continuous was understood as a quantity that could be divided infinitely, leading to the philosophical debate about whether it could be composed of “indivisible” points or infinitesimals. While classical mathematics eventually replaced infinitesimals with the limit-based definition of continuity, modern mathematical approaches like non-standard analysis have revived the use of infinitesimals to provide a more intuitive foundation for calculus.

Mathematical perspective

Classical vs. modern definitions: Historically, a function was considered continuous if an infinitesimal change in the input resulted in an infinitesimal change in the output. However, the 19th-century abandonment of infinitesimals led to the more precise limit-based definition still used today.
Infinitesimals and limits: The classical notion of infinitesimals was based on a non-zero quantity closer to zero than any other non-zero real number. While this concept was formally replaced, it laid the groundwork for calculus, which was developed by mathematicians like Fermat using the “infinitesimal variation” technique to find tangents and maxima/minima.
Non-standard analysis: This modern branch of mathematics provides a way to work with infinitesimals in a rigorous framework, which can be more intuitive for some applications than the limit-based approach.

Philosophical perspective

The nature of the continuum: Philosophers have long debated the nature of continuous quantities. Some, like Aristotle, argued that a continuum could not be composed of points (indivisible units). Others, like the Stoics, maintained that space, time, and matter were continuous.
Atomism vs. continuity: The debate also involved atomists, who believed in indivisible units of matter. The opposing view was that the continuum is a unified whole, which can be “divided” indefinitely without ever reaching a true “first” or “last” part.
Reduction of the continuous to the discrete: A major theme in the history of both mathematics and philosophy has been the attempt to reduce the continuous to the discrete, a process that has been pursued by mathematicians like Bolzano and Cauchy.

AI responses may include mistakes.

[1] https://www.amazon.com/Continuous-Discrete-Infinitesimal-Philosophy-Mathematics/dp/3030187098

[2] https://digitalcollections.dordt.edu/cgi/viewcontent.cgi?article=2150&context=faculty_work

[3] https://dokumen.pub/the-continuous-the-discrete-and-the-infinitesimal-in-philosophy-and-mathematics-1st-ed-2019-978-3-030-18706-4-978-3-030-18707-1.html

[4] https://plato.stanford.edu/archives/win2018/entries/continuity/

[5] https://plato.stanford.edu/archives/sum2013/entries/continuity/

[6] wikipedia/en/Infinitesimal