Is a number actually a quantity, or just a ghost of one?
In short: We often view numbers as the core of reality, but their history reveals a complex evolution. From Aristotle's ancient categories to Newton's abstract ratios, the definition of quantity has shifted from physical magnitudes to mathematical structures, allowing us to transform subjective qualities into the measurable data that powers modern science.
In ancient Greek philosophy, specifically the ontology of Aristotle, quantity was viewed as a fundamental scientific category. Aristotle distinguished between two primary types: 'quantum' and 'magnitude.' A quantum is a plurality—something divisible into discrete, countable parts like an army or a flock. If it is numerable, it is a plurality; if it is measurable, it is a magnitude. This distinction separates the 'how many' (multitude) from the 'how much' (magnitude).
The transition from physical magnitude to abstract number is a pivotal moment in mathematical history. Whole numbers served as the conceptual basis for relations according to Euclid and Aristotle. However, John Wallis later reimagined ratios of magnitudes as real numbers. Newton formalised this transition, viewing numbers as abstract ratios between similar quantities rather than mere collections of units. Essentially, the ratio of any magnitude—whether it be volume, mass, or heat—results in a number, stripping away the original 'genus' of the quantity.
Modern science relies on this ability to render abstract qualities into physical quantities. Physics, in particular, is a quantitative science because it postulates that all material bodies with physical dimensions are subject to measurement. This involves distinguishing between intensive quantities, like density and pressure, which do not depend on the size of the system, and extensive quantities, like mass and energy, which are additive. By establishing these quantitative structures, we move from the subjective 'quality' of an object to the objective, measurable 'quantity' that forms the cornerstone of modern scientific progress.
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- Summary by Intellect · source: Quantity
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