Quantity
Quantity or amount is a property that includes numbers and quantifiable phenomena such as mass, time, distance, heat, angle, and information. Quantities can commonly be compared in terms of “more”, “less”, or “equal”, or by assigning a numerical value multiple of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others function as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little.
Under the name of multitude comes what is discontinuous and discrete and divisible ultimately into indivisibles, such as: army, fleet, flock, government, company, party, people, mess (military), chorus, crowd, and number; all which are cases of collective nouns. Under the name of magnitude comes what is continuous and unified and divisible only into smaller divisibles, such as: matter, mass, energy, liquid, material—all cases of non-collective nouns.
Along with analyzing its nature and classification, the issues of quantity involve such closely related topics as dimensionality, equality, proportion, the measurements of quantities, the units of measurements, number and numbering systems, the types of numbers and their relations to each other as numerical ratios.
Across various philosophical traditions, the concept of “quantity” is explored using different terms and distinctions, reflecting underlying metaphysical and epistemological perspectives. [1, 2]
Ancient Greek Philosophy
- πoσóv (poson): The primary term used by Aristotle in his Categories for “quantity”. He defined it as that which is divisible into constituent parts.
- Plurality/Number: Refers to discrete quantity (countable, individual units), such as a certain number of men or books.
- Magnitude/Extension: Refers to continuous quantity (measurable, infinitely divisible parts with common boundaries), such as a line, surface, volume, space, or time.
- Forms (Platonic tradition): For Plato, numbers and quantities were not mere properties of physical objects but existed as perfect, abstract Forms in an independent realm, in which physical things imperfectly “participated”. [1, 2, 3, 4, 5]
Medieval Philosophy
- Quantitas: The Latin term for quantity, widely used by Scholastic philosophers like Thomas Aquinas, who followed Aristotle’s distinction between discrete and continuous quantity.
- Accident: Quantity was generally considered an “accident” of a substance, meaning it is a property that can change without the substance itself ceasing to exist (e.g., a lump of clay can change shape and size but remain clay).
- Multitude: Used to specifically refer to discrete quantity or number.
- Extension/Magnitude: Used for continuous quantity, particularly in debates about whether matter could exist without being spatially extended. [4, 6, 7]
Modern Philosophy
- Extension: René Descartes identified extension (length, breadth, and depth) as the primary and essential attribute of physical matter, effectively making spatial quantity foundational to the physical world.
- Primary Qualities: John Locke used this term for qualities like solidity, extension, figure, motion, and number, which he argued were inherent in objects themselves and existed independently of human perception.
- Categories of Understanding: Immanuel Kant, in his Critique of Pure Reason, identified quantity (specifically unity, plurality, and totality) as one of the fundamental, innate structures or “categories” of the human mind, which organize our sensory experience rather than being objective properties of “things-in-themselves”.
- Quantum: Hegel used “quantum” to refer to a determinate or limited quantity, a specific stage in the dialectical development of the concept of quantity.
- Monads (Leibniz): Gottfried Leibniz proposed a universe composed of non-extended, qualitative monads, arguing that quantity and extension were not primary qualities but rather well-founded phenomena or appearances arising from their aggregation. [1, 4, 8, 9, 10, 11]
AI responses may include mistakes.
[1] https://www.planksip.org/the-philosophical-problem-of-quantity-and-philosophy-1761722338376/
[2] https://www.planksip.org/the-philosophical-problem-of-quantity-and-philosophy-1761353058659/
[3] https://www.planksip.org/the-logic-of-quantity-and-relation-and-logic-1761683614721/
[4] https://www.planksip.org/the-philosophical-problem-of-quantity-and-philosophy-1761208569673/
[6] https://www.planksip.org/the-philosophical-concept-of-number-quantity-and-philosophy-1760283074274/
[8] https://www.marxists.org/reference/archive/hegel/works/sl/slquant.htm
[9] https://plato.stanford.edu/entries/categories/
[10] https://www.planksip.org/the-logic-of-quantity-and-relation-and-logic-1761261586032/
[11] https://www.marxists.org/reference/archive/hegel/works/sl/slquant.htm
Philosophical quantities are primarily distinguished between discrete and continuous types. Discrete quantities are countable and composed of separate units, like the number of books or individual atoms. Continuous quantities are measurable and infinitely divisible, such as length, time, or space. Other distinctions include extrinsic quantity, which depends on other objects, and intrinsic quantity, which is an object’s inherent property. [1, 2, 3]
Discrete and Continuous Quantities
- Discrete Quantity: Composed of distinct, separate units that can be counted.
- Examples: A number of planets, individual atoms, or books on a shelf.
- Key Issue: The philosophical problem of infinitesimals, as each unit is seen as indivisible.
- Continuous Quantity: Measurable and divisible into infinitely smaller parts without losing its nature.
- Examples: Length, time, space, or volume.
- Key Issue: Zeno’s paradoxes, which question how motion is possible if a distance can be divided into an infinite number of points. [1, 2, 4]
Intrinsic and Extrinsic Quantities
- Intrinsic Quantity: A property inherent to an object itself, independent of external relations.
- Example: The mass of an object.
- Extrinsic Quantity: An attribute that depends on an object’s relationship to other things or a standard.
- Examples: An object’s weight (relative to gravity) or speed (distance over time). [3]
AI responses may include mistakes.
[1] https://www.planksip.org/the-philosophical-problem-of-quantity-and-philosophy-1760257826268/
[2] https://www.planksip.org/the-philosophical-problem-of-quantity-and-philosophy-1761353058659/
[3] https://www.planksip.org/the-philosophical-concept-of-number-quantity-and-philosophy-1760283074274/
[4] https://www.planksip.org/the-philosophical-problem-of-quantity-and-philosophy-1761612469320/
Incommensurable quantities are two quantities of the same type that cannot be measured by a common unit, meaning their ratio cannot be expressed as a ratio of whole numbers. The classic example is the side and the diagonal of a square, as their lengths are incommensurable, and their ratio is the irrational number 2√. Another example is the radius and circumference of a circle, because their ratio is 𝜋, which is also an irrational number.
Key characteristics
- Lack of a common measure: Incommensurable quantities do not share a common unit that can measure both exactly an integral number of times.
- Irrational ratio: The ratio of two incommensurable quantities is always an irrational number.
- Historical significance: The discovery of incommensurable quantities, like the side and diagonal of a square, was a major development in ancient Greek mathematics (Pythagorean and Euclidean) that challenged the idea that all quantities could be expressed as whole number ratios.
Examples
- Side and diagonal of a square: The ratio of the diagonal to the side is 2√, an irrational number.
- Radius and circumference of a circle: The ratio of the circumference to the radius is 2𝜋, which is irrational.
- Numbers with no common factor: In number theory, two integers are sometimes called incommensurable if they have no common divisor other than 1, though this usage is less common than the geometric definition.
Contrast with commensurable quantities
- Commensurable quantities can be measured by the same unit, and their ratio is a rational number.
- For example, a foot and a yard are commensurable, as they can both be measured in inches, and their ratio is (1:3)(a rational number).
AI responses may include mistakes.
[2] https://i2insights.org/2019/04/09/understanding-incommensurability/
[3] https://www.youtube.com/shorts/yZWmjAMPBTs
[5] https://www.britannica.com/science/incommensurable
[6] https://en.wiktionary.org/wiki/incommensurable
[7] https://encyclopediaofmath.org/wiki/Commensurable_and_incommensurable_magnitudes
[8] https://mathworld.wolfram.com/Incommensurate.html
[9] https://webstersdictionary1828.com/Dictionary/incommensurable