Measurement and State
Modern Logic of Measurement
In mathematics, and in science, measurement is defined as the process of assigning a numerical value to an attribute or characteristic of an object or event. This numerical value allows us to compare that attribute with other objects or events. How tall is this table? How heavy is that book? How long did it take to run that race? The key to this process is comparison. This assignment of a numerical value is done by comparing the attribute of the object to a standard unit.
So, what exactly is a standard unit?
A standard unit or simply a unit is the reference, the benchmark, the agreed-upon quantity we use to perform a measurement. It’s the ”one” that everything else is compared against. For example, a metre is a standard unit of length. A kilogram is a standard unit of mass. A second is a standard unit of time. And what is the measurement itself?
The measurement is the result you get after comparing an object to that standard unit. If a table is ”2 metres long,” the ”2 metres” is the measurement. It’s the numerical value and the unit, describing the table’s length.
Now, the logical deduction of the above definition is quite interesting:
Measurement is a process of assigning a numerical value to an attribute.
This assignment is done by comparing the attribute to a standard unit.
A standard unit is the reference we use to perform a measurement.
According to Point 3, if a standard unit is what we use to perform a measurement, what does that imply about its existence relative to the measurement?
It implies that the standard unit must exist before the measurement can take place.
Something in metres cannot be measured if the concept of a metre doesn’t already exist as a defined standard. The metre is the tool, the ruler, the benchmark. So, if a metre is a standard unit, and a standard unit must exist before any measurement is made using it, then logically, a metre itself cannot be a measurement.
Let’s clarify this with an example: We often see ”1 m” and think of it as a measurement. But in the context of the above deduction, ”1 m” isn’t a measurement of an object. It’s the definition of the standard unit itself. When we say ”1 m”, we are referring to the standard unit that we then use to measure other things. If a statement says ”this stick is 1 m long,” then ”1 m” is a measurement of the stick. But if someone just points to the definition of a metre and says ”this is 1 m,” he is referring to the unit, not a measurement of an object. The distinction is subtle but crucial. The standard unit is the tool or reference; the measurement is the result obtained by using that tool on something else. Here is the logical flow given below:
Measurement is assigning a number by comparison.
The comparison is against a standard unit.
The standard unit exists before the measurement.
Therefore, the standard unit itself is not a measurement, but the means by which measurements are made.
Hence, the conclusion can be made that “1 metre is not a measurement!”
This raises a significant concern and demands a more robust and holistic definition of measurement.
Māpana (Measurement)
A discussed above, measurement in mathematics is often not well understood. This framework can give a new perspective on the definition of measurement making it more robust from earlier.
The measurement is the process of observing and quantifying पदार्थ (Padārtha or Quantity) by performing certain operations or actions on it, guided by the chosen मात्रा (Mātrā or Unit).
The most fundamental measurement is called as a मात्रा (Mātrā or Unit), which is used for comparison and quantification on that scale.
In short and simple words, measurement is all about observation, quantification and comparison of पदार्थ (Padārtha). This action-oriented definition is the cornerstone of this thesis, as it is the mechanism by which numbers emerge from a physical process.
But, how to quantify a quantity? Before delving into the understanding and definition of measurements, it is important to comprehend the concept of "Quantity" in mathematics.
Quantity
The conventional English definition of a quantity as, “an amount, magnitude, or number that can be measured or counted,” focuses primarily on numerical value. This perspective, however, proves insufficient for the broader ontological framework required for this study, which must account for the nature of knowable objects before their quantification. The comprehensive system of categories articulated in classical Indian philosophy offers a more robust foundation.
The तर्कसंग्र: (Tarkasaṅgraha), meaning “A Compendium of Reasoning,” was authored by Annaṁbhaṭṭa, a 17th-century scholar from the Andhra region of India. It is a prakaraṇa grantha, a topic-specific manual designed to serve as a concise introduction for students. Its primary goal is to synthesize the philosophies of two major schools of Hindu thought:
न्याय (Nyāya): The school of logic and epistemology (the study of knowledge).
वैशेषिक (Vaiśeṣika): The school of metaphysics and ontology (the study of reality and existence).
By blending these, Annaṁbhaṭṭa provides a complete and systematic framework for understanding reality and the means of knowing it. On the basis of the rigorous classificatory principles and ideas found in the तर्कसंग्र: (Tarkasaṅgraha), this framework is redefining and expanding the definition of the term “Quantity”, here termed as पदार्थ (Padārtha or Quantity). This definition, which will be used exclusively within the context of this research, is not limited to mere magnitude but extends to the fundamental nature of any entity that can be subject to observation and, ultimately, to a mathematical formalism.
A पदार्थ (Padārtha or Quantity) is defined as any tangible or intangible entity, property, characteristic, or abstraction that can be the subject of observation and potential measurement.
It is the “mathematical object” in its pre-quantified state. To delineate the scope of what is measurable, quantities are classified into four distinct categories:
तत्व (Tattva): A quantifiable entity that possesses properties. This category includes discrete objects like apples or continuous substances like water. However, it is uncertain whether every तत्व (Tattva) in its substratum is discrete or continuous. The possibility of it being continuous cannot be dismissed.
परतत्व (Paratattva): A quantifiable entity that possesses properties but abstract in nature. This category includes abstract objects like geometrical shapes, vectors, matrices, etc.
गुण (Guna): A property itself, either of तत्व (Tattva) or परतत्व (Paratattva), which can also be quantified, such as redness or sweetness. While one can measure the amount of redness, “redness” as an abstract quality is not directly measured. Love, coldness, angle, length, etc. comes under this category.
प्रकृति (Prakṛti): A foundational substratum that is unquantifiable and possesses no properties. This represents the underlying reality from which quantifiable entities emerge, a concept beyond the scope of direct or indirect measurement.
This framework applies only in the realm of तत्व (Tattva), परतत्व (Paratattva) and गुण (Guna), upon which the action of measurement can be performed. प्रकृति (Prakṛti) needs an entirely different framework which should be different from human reasonings and logics because only when प्रकृति (Prakṛti) manifests, results in the existence of other three पदार्थ (Padārtha or Quantity).
What is a “System”?
A system here represents the following elements: Your measurement space, quantified (unit), the behaviour of the quantified and the observer performing actions. All these elements together make up a system what we call as the “Measurement System”.
State of a System
A state is like a final snapshot of the measurement system. If the final measurement mimics another state, then the measurement must be equal too. Consider a space with nothing in it. Now, an apple appears within this space. This event happens once more, and then one of the apples is removed. What observations can be made about this sequence? What did you observe?
Last updated