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Dmytro Taranovsky

March 31, 2000

Last Modified:
May 5, 2002

All information about the external world comes to you through your perception. Therefore, your perception is your only source of information about the outside world. For example, when you measure temperature using a thermometer, your eyes obtain the information while the thermometer is only a tool.

It is built in to our minds that the world is reasonable and has
patterns in it. More precisely: **Postulate**: Every possible
pattern has a reasonable probability to be valid in any particular
system.

In other words, if a property was observed every time an observation was made for sufficiently many observations, people would assume that the property would be observed the next time the observation is made. While reasonable probability can be very small for some complex patterns, even unusual patterns can be observed. The example below illustrates how the scientific method is derived from the postulate.

**Data:** In a certain field, two competing theories exist.
Initially, the first theory is believed to be correct with
probability **p _{0}**. An experiment is run. According to
the first theory, the results would match the obtained results with
probability

Since you perception is your only interface with the rest of the world, a theory can be tested only on what perception it predicts for you. Therefore, everything else must be left to religion. Theories give predictions by using a (mathematical) model for the world or an aspect of the world and relating the model to human perception. Theories do not state the reality: They state models. Two theories that predict the same perceptions in all cases are scientifically (and mathematically) equivalent regardless of the model they use even if one theory is merely the expression of another as relations between feelings without any reference to the external world. As part of our common religion, the external world exists and every human with at least a minimum level of intelligence is an observer. Since theories can only relate different human perceptions, they cannot be used to explain why humans can feel.

**Example:** The theory of evolution, which is thoroughly
confirmed by experiments, uses a model stating that all life
including humans descended from inorganic substances over the last
several billion years. The model is used for (among other things)
predicting the anatomy and physiology of different species and
predicting which fossils are likely to be found. The predictions are
valid according to the Postulate (since so many predictions of the
theory are confirmed, another prediction should be true as well) and
should be used. Thus, the model should be taught.

It is part of
your religion, however, whether humans evolved from monkeys or
whether God created the world in such a way as to make it appear that
humans evolved from monkeys. Many people do not accept the model as
the reality. Unfortunately, people did not understand that descending
of humans from animals could be considered a model. Thus, those whose
religion did not allow them to accept the model as reality ignored
the theory and wasted billions of dollars.

To simplify the state of mind, it is often convenient to temporarily think of models as reality. For example, when people play computer games, they temporarily think that the virtual world portrayed there is real.

Usually, physical models use physical quantities. All physical quantities must be defined to be used. Many are defined as real numbers[1]. A physical quantity is called measurable if it can be evaluated experimentally. All measurable quantities measure an aspect of the world. To facilitate making and understanding of models, many measurable quantities are defined through measurement, so that the quantities would be the same in different models. Immeasurable quantities may or may not measure an aspect of the world depending on your religion. For example, according to the theory of relativity, distance without a frame of reference is an immeasurable quantity. However, it is intuitive that such quantity as distance exists.

**Example:** Our world is well modeled by the four-dimensional
Euclidian space-time: the set of all ordered quadruples of real
numbers (x, y, z, t). Distance between (x_{1}, y_{1},
z_{1}, t_{1}) and (x_{2}, y_{2}, z_{2},
t_{1}) is defined as sqrt((x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}).
Position in space (denoted by **r**) of (x, y, z, t) is (x, y, z).

A model that leaves some physical laws unspecified is called a template model. The study of template models is called mechanics[2]. Template models often include conservation laws (some quantities are conserved) and relativity principles (physical laws are invariant under some sets of transformations). These are very helpful in specifying models. When a model is fully specified, these are useful theorems.

**Example:** An important conservation law is the conservation
of momentum. Momentum of a system is defined, informally, as the sum
of products of mass and velocity for each object.

**Example:** A powerful principle of relativity is the
invariance of physical laws under all continuously accomplished
transformations of coordinates that leave
(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}-(t_{2}-t_{1})^{2}
between any 2 points (x_{1}, y_{1}, z_{1},
t_{1}) and (x_{2}, y_{2}, z_{2}, t_{2})
invariant (not changing).[3] A
transformation is called continuously accomplished if there is a
continuous set of continuous intermediate states starting with the
initial state (before the transformation) and ending with the final
(after the transformation).

In most cases, the ratio of a quantity to the given quantity (the
value of a given quantity is called a unit) is important. The
quantity is then expressed as a product of the ratio and the
unit.**Example:** 2s is 2*s (second (s) is SI unit for time).

Units are usually defined as real numbers (and manipulated
accordingly) even if they characterize a vector.**Example:**
(1, 2, 3) m = (1m, 2m, 3m) (meter (m) is SI unit for distance).

A useful concept in physical science is dimension. This concept is
best introduced from the historical point of view. Long ago, physical
quantities such as distance seemed unknown and mysterious. However,
the ratio of length to some specified length (such as one meter) is a
real number and can be manipulated. Therefore, quantities were
expressed as a product of a unit and a value. This happened to the
basic quantities (length, time, mass, electric charge, temperature,
and the amount of substance). Other physical quantities, called
derived, are defined from the basic. For example, velocity is defined
as d**r**/dt [4] (that is the
rate of change of position). Since the value of the basic units is
undefined, any formula that defines a basic unit in terms of other
basic units is incorrect. Similarly, if all formulas that define a
basic unit from a quantity not containing the unit are incorrect, the
unit is not defined. Therefore, every physical quantity has units for
every basic quantity raised to the unique power. (Otherwise, an
equation to determine a basic unit could be set up.) These powers
constitute the dimension of a quantity. For example, force has units
of mass^{1}*length^{1}*time^{-2}*charge^{0}*temperature^{0}*(amount
of substance)^{0} and dimension (1, 1, -2, 0, 0, 0). If in an
equation these powers (that is dimension) would be different, the
equation has a mistake. Dimensions are also helpful in recalling
formulas since the incorrect versions are often dimensionally
incorrect. Checking that dimensions are correct is called dimensional
analysis.

As stated above, all physical quantities must be defined. The best
way to define the quantities is to assign a value for each basic
unit. The values are assigned to make the most fundamental values
equal to one. The most fundamental constants (values) in physics are
**c** (the speed of light), **G** (the gravitational constant),
**ℏ
** (Planck's constant (h-bar), ℏ

Values are converted between different units by multiplying the
values by a combination of the constants and computing the results
using the value of the constants in the traditional system of
units.**Example:** c=299792458m/s. The speed of 1/14 is
1/14*c=21413747m/s.

**Example:** Proper distance between (x_{1}, y_{1},
z_{1}, t_{1}) and (x_{2}, y_{2}, z_{2},
t_{2}), sqrt((x_{2}-x_{1})^{2} +
(y_{2}-y_{1})^{2} + (z_{2}-z_{1})^{2}
- (t_{2}-t_{1})^{2}), was written with c^{2}
preceding (t_{2}-t_{1})^{2} since c was not
considered to be equal 1.

If you want to obtain the full benefit of dimensional analysis, use the names of the constants or their traditional values. Using only some of the names will give some benefit of dimensional analysis though not the full benefit. On the other side, using '1' instead of the fundamental constants can make the formulas much more elegant and simpler and can greatly simplify the numerical calculations. From the practical point of view, defining physical quantities offers additional, often beneficial options and is thus good.

**Example:** Light falls onto a surface perpendicular to the
light. The light flux was traditionally characterized by 4 different
quantities: relativistic mass-density (M/V=M/(A*d)), energy density
(E/V=E/(A*d) (E=M*c^{2})), energy flux (P/A=E/(At) (t=d/c)),
and mass flux (M/(At)). In reality, the quantities are the same:
Energy (E) is relativistic mass (M) and d=t (the distance traveled by
light is equal to the time traveled).

A system of units (where each quantity is assigned a unit) under a
set of formulas is called consistent if the results would be
unaltered when all units are removed from the data (that is when each
quantity is divided by the appropriate unit) and added only to the
results. If no special restrictions are placed, a system of units is
generally inconsistent since not all units are equal to 1: The value
with units is different from the value with units removed. To make a
system consistent, special conversion constants are added. These
constants must be used (if the system is to remain consistent) when
relating a value with one unit and a value with a different unit. In
the traditional (SI) system of units, the constants are c, G, ℏ_{A}. The value of the conversion constants is 1;
however, their values divided by the SI units for them are not 1
making multiplication by them an equivalent of a conversion from one
unit to another. If a system of units is consistent, dimensional
analysis can be applied. The number of basic quantities is the number
of conversion constants. (Each basic unit needs a conversion constant
(to be related to other units), and a conversion constant is not
needed when all basic units can be converted with the existing
constants.)

**Example:** Energy of a photon (E) is equal to its angular
frequency (*w*). However, in SI units a conversion constant (ℏ*ℏ
w*.

For each set of problems, you have a choice on whether to use the conversion constants. To simplify the calculations, if you are using a system of units that is consistent for the formulas you are using, you may drop off the units and add them only to your result. However, if you do so, you must state it explicitly; otherwise, your solution would be wrong since it would have an incorrect statement such as 5kg=5 (kilogram (kg) is SI unit for mass). By choosing the units and the conversion constants, you can customize the dimensional analysis to make it most beneficial for your problem.

[1] As will be explained later, this was not always the case. Real numbers are also defined. Their definition can be found in many algebra textbooks.

[2] Some scientists prefer a different but similar definition of mechanics.

[3] The transformation is discussed in any text on special relativity. Traditionally, 'c' preceded 't' for the reasons described below.

[4] Recall that **r** is
position, d**r**/dt=d(x, y, z)/dt=(dx/dt, dy/dt, dz/dt);
dx/dt=limit as t_{1}→_{1})-x(t))/(t_{1}-t).