Functionsas Physical Dimensions

Ingvar Johansson

Informal symposium on philosophy of biology

Buffalo, NY, October 15, 2006

Starter Slide

• Pre-modern Aristotelians: Mathematics cannot be especially useful in the sublunary world.

• Galileo Galilei: The book of nature is written in the language of mathematics.

• Both are wrong: Some parts of nature, but not the whole of it, can be represented in terms of quantities.

International System of Units

• The International System of Units (abbreviated SI from the French language name Système international d'unités) is the modern form of the metric system. It is the world's most widely used system of units, both in everyday commerce and in science. (Wikipedia)

7 Base Quantity Dimensions

Base and Derived Units

Units based on undefined SI dimensions: meter, second, kilogram, ampere, candela, kelvin, mole.

Units based on defined SI dimensions: volume, area, velocity, acceleration, newton, joule, pascal, coulomb, farad, henry, hertz, lumen, lux, ohm, etc.

Dimensions can be multiplied and divided (v = m/s).

The Units are Conventional

The Dimensions are Not!

• All specific lengths can be compared with another, but no specific length can be compared with a quantity from any other dimension.

• And this is not so because we have constructed the dimensions this way.

Laws for Dimensions

• It makes no sense to say that 5kg is more than, or longer than, 4m.

• The addition 5kg + 4m makes no sense.

• A thing can be both 5kg and 4m, but it cannot possibly at one and the same time be both 5kg and 4kg,nor both 5m and 4m.

Quantities Contain a Duality

• It is impossible to understand scales and additions if one does not accept that each determinate quantity (e.g.,5kg, 4m) contains a duality:

(1) a physical dimension (mass, length) that it shares with other determinate quantities, and

(2) a unique determinateness that is referred to by means of the numeral (5, 4).

• A physical dimension might also be called a determinable.

Quantities are ‘Repeatables’

• Many different things can simultaneously have a mass of 5kg (length of 4m, etc.).

• Determinate quantities are ‘repeatables’ or universals.

• When a determinate quantity is repeated, its dimension is repeated, too.

Speech Acts Using Quantities

• Every true statement such as ‘T has a length of 4m’, ‘T has a mass of 5kg’, and ‘T has a temperature of 253K’ refers explicitly to one determinate quantity of a physical dimension, but it contains an implicit reference to all the determinate quantities of this dimension.

• This makes comparisons easy.

My proposal

• There are at least 9 Base Physical Dimensions.

• In particular, there is the dimension ‘Function’ that subsumes many functions, each of which takes degrees of functioning.

7 Base Quantity Dimensions

9 Base Physical Dimensions

Dimension Unit Symbol

1. Length meter m

2. Mass kilogram kg

3. …

7. Amount of mole mol

Substance

8. Shape --- ---

9. Function degree-of-functioning ---

Undefinables

• The Base Dimensions are Undefined.

• On pain of an infinite regress, there have to be undefined terms, concepts, and ‘repeatables’ (universals).

• If one turns a base SI unit into a derived unit, then some formerly derived units become base units.

• The dimension ‘Function’ is undefinable.

Undefinables are Not Mystical

• Undefinable words, terms, and concepts can be learned and taught.

• They can even be given verbal characterizations.

• Ostensive teaching is not the same as giving ostensive definitions.

• So-called ‘implicit definitions’ supply us with one kind of undefinables.

The Base Dimension of Function

a) Function is, unlike all seven SI dimensions but like Shape, a non-quantitative physical dimension.

b) Specific functions take (in their functioning), like all seven SI dimensions, degrees.

c) Functions cannot be assigned a measuring unit, but each function can be assigned a prototypical functioning.

d) Function is, like the SI dimension Amount of Substance, a dimension that has to be specified before it becomes a true dimension.

The Base Dimension of Function

a) Function is, unlike all seven SI dimensions but like Shape, a non-quantitative physical dimension.

b) Specific functions take (in their functioning), like all seven SI dimensions, degrees.

c) Functions cannot be assigned a measuring unit, but each function can be assigned a prototypical functioning.

d) Function is, like the SI dimension Amount of Substance, a dimension that has to be specified before it becomes a true dimension.

Like the SI Units, Shapes are ‘Repeatables’ that Contain a

Duality

• A circle is a shape; a square is shape;this is a shape; this is a shape.

• The same determinate shape can exist in many places simultaneously.

• Shapes cannot be compared with any of the seven SI quantities.

Scales and Similarity Relations

0 1 2 3 4 5 6 7 8 9 10 cm

7 cm is more like 4 cm in length than 1 cm.

Scales and Prototypes: Shapes (1)

• There is a scale for all ellipses that have one axis in common.

• Each determinate ellipse represents a certain ‘eccentricity’.

Scales and Prototypes: Shapes (2)

• There is simply no scale or other metric for all shapes.

• Some dimensions can because of their nature not be quantified, not even ordered.

• But there are nonetheless many resemblance relations even between shapes.

Scales and Prototypes: Shapes (3)

• Prototypical representation is possible for a larger set of shapes.

• Prototypes require a dimension (e.g.,Shape); and perhaps sub-dimensions.

• Similarity is always similarity in a certain respect.

Scales and Prototypes: Color Hues

• Color hues can be represented both by means of scales (‘logical classification’) and prototypical classification.

Typical: blue green y red

Two Neglected Views

1. The physical dimension of shape cannot because of its nature be quantified, nor even ordinally scaled.

2. What the use of scale units are to quantified dimensions, the use of prototypical concepts and descriptions can be to non-quantified dimensions.

Scales and Prototypes

• Prototypical conceptualizations can be just as scientific as

scale constructions are.

The Base Dimension of Function

a) Function is, unlike all seven SI dimensions but like Shape, a non-quantitative physical dimension.

b) Specific functions take (in their functioning), like all seven SI dimensions, degrees.

c) Functions cannot be assigned a measuring unit, but each function can be assigned a prototypical functioning.

d) Function is, like the SI dimension Amount of Substance, a dimension that has to be specified before it becomes a true dimension.

Functions and Functionings

• A living heart has a function. Normally, it is in an actual state of functioning.

• If it is outside a body, as in a heart transplantation, it has its state of functioning only potentially, or as a dispositional property.

• In its state of functioning, the heart participates in a process, and performs a certain characteristic movement.

Functionings Take Degrees

• A functioning function has always a certain degree of well-functioning or malfunctioning.

• As (e.g.) Length is necessarily connected with determinate lengths, Function is necessarily connected with degrees of functioning.

The Base Dimension of Function

a) Function is, unlike all seven SI dimensions but like Shape, a non-quantitative physical dimension.

b) Specific functions take (in their functioning), like all seven SI dimensions, degrees.

c) Functions cannot be assigned a measuring unit, but each function can be assigned a prototypical functioning.

d) Function is, like the SI dimension Amount of Substance, a dimension that has to be specified before it becomes a true dimension.

Prototypes for Functioning

• Prototypical representation is possible for a large set of heart functionings.

• Prototypes require a dimension (e.g., Function ); and perhaps sub-dimensions.

• Similarity is always similarity in a certain respect.

The Base Dimension of Function

a) Function is, unlike all seven SI dimensions but like Shape, a non-quantitative physical dimension.

b) Specific functions take (in their functioning), like all seven SI dimensions, degrees.

c) Functions cannot be assigned a measuring unit, but each function can be assigned a prototypical functioning.

d) Function is, like the SI dimension Amount of Substance, a dimension that has to be specified before it becomes a true dimension.

Amount of Substance

• What is a mole?

• It is simply a number such as dozen (12) or gross (144). It is: 6.022 x 10-23.

• “When the mole is used , the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles or specified groups of such particles.” (Resolution 3 of the 14th CGPM, 1971)

Pre-History: Amedeo Avogadro

• Gases can independently of kind always be compared with respect to Amount of Substance.

• Avogadro’s Law: Equal volumes of gases, at the same pressure and temperature, contain the same number of molecules.

• Avogadro’s Number: The number of molecules in a mole (6.022 x 10-23).

• This is the number of atoms in 0.012 kilogram of carbon-12.

Some True Base Dimensionsconnected to

Amount of Substance

number of molecules of kind X

number of ions of kind X

number of atoms of kind X

number of fermions of kind X

number of bosons of kind X

Speech Acts Using Quantities

• Every true statement such as ‘T has a length of 4m’, ‘T has a mass of 5kg’, and ‘T has a temperature of 253K’ refers explicitly to one determinate quantity of a physical dimension, but it contains an implicit reference to all the determinate quantities of this dimension.

• This makes comparisons easy.

Speech Acts Using Functions

• Every true statement such as ‘the function of the heart is to pump blood’, ‘the function of x is to F’, refers explicitly to one prototypical functioning of a specification of the physical dimension Function, but it contains an implicit reference to all the corresponding degrees of functioning.

• This makes comparisons easy.

Base Physical Dimensions

Non-Quantitative Quantitative

Functions Shapes Amount-Sub Qualities molecules-kind-X length

ions-kind-X mass atoms-kind-X time fermions-kind-X el. current bosons-kind-X temperature

lum. intensity

The EndThe End