Mathemematics:Perspectives on the Nature of

Space, Time, Humanity, and God

Erik ScottHighline Community College

Math and humanity or God?That ain’t the math I know!!

Key Question I:What is the shape of space?

Relevant Mathematical Theme

We can distinguish some shapes from others simply through measurement of particular local properties.

Implication:

There are ways to reach conclusions about the shape of the universe without having to

“look at it from the outside.”

A “Down to Earth” Example

What you see:

What it could be:

A “ in the Sky” Picture of the Universe

If we only use information obtained locally(our “neighborhood” in math-speak)

the conclusions we can draw are limited but helpful.

“Flat” spaceHyperbolic space

Spherical space

For help, we turn to our trusted assistant: the triangle.

Key Question II:What is “time”?

Relevant Mathematical Theme

Decisions to prioritize or isolate variables strongly influence our perception of relationships between them.

Implication:

We tend to view time as an independent “thing” which we use to measure events or put them in an

order. Could this decision be limiting our ability to truly understand the nature of time?

Einstein’s Legacy:Absolute time does not exist

This, combined with the fundamental relationship

Distance = Speed x Time

ultimately led Einstein to conclude that events occurring at nearly the speed of light can be observed as taking different lengths of time.

Einstein postulated that the speed of light is the same for all observers, regardless of whether the light

source or observer is also moving.

Two Perspectives on Time

A view of an event in which time is simply way to organize or reference individual steps.

A view of an event in which time is intimately connected to the physical action of the event.

(A curve in space-time.)

Key Question III:How is human thought restricted?

Relevant Mathematical Themes

The more variables we include in our models of reality, the more complex our models and reasoning become.

– AND –

Certain ideas cannot be discovered or proven true,

regardless of the thinker’s ingenuity.

A Mathematical Metaphor forHow We Model Reality

A theory usually begins with observation of reality, so theory and

reality agree on at least that data.

But as the theory is applied to situations further from the original, it

must flex to include the new data

0for 1,,,, zyxzyx

1,0,0,,for 1,1

2,

1

2,,

zyx

z

y

z

xzyx

In fact, when the metaphor of representing a sphere by points on a plane is turned into mathematical reality, you can see the rise in complexity just described.

If we only try to capture the points on the upper half of the sphere, the math looks like:

But to capture nearly all of the sphere, we get:

A Mathematical Metaphor forHow We Model Reality

That Which We Cannot Know:Gödel's Incompleteness Theorem

EVERY logical system complex enough to support the infinite set of counting numbers will lead to true statements that cannot be

proven within the same logical system.

Implications:

The more knowledge we gain, the more questions we will raise, and some of those

will indeed be unanswerable.

Key Question IV:Does math support beliefs about God?

Relevant Mathematical Themes• One cannot use a finite process to reach the

infinite.

• Given a subspace of a higher-dimensional space, it may be impossible to recognize the existence of the higher-dimensional space.

• If one knows all the forces acting on an object, and exactly what the object is doing at one instant in time, then both the history and future of the object can be predicted.

Key Question V:How can all of this stuff be “math”?

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22

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Mathematics is the Study of Patterns

• Algebra – Number patterns

• Geometry – Patterns in shapes

• Calculus – Patterns of motion

• Logic – Patterns in reasoning

• Probability – Randomly-generated patterns

To learn more, read:

Carl Boyer – A History of MathematicsFritjof Capra – The Tao of PhysicsA.K. Dewdney – The PlaniverseStephen Hawking – The Universe in a NutshellDouglas Hofstadter – Gödel, Escher, Bach: The Eternal

Golden BraidBertrand Russell – Introduction to Mathematical

PhilosophyJeffrey Weeks – The Shape of Space