Fundamental questions are often questions of scale: What is
the smallest sustainable flame? What is the minimum bubble diameter? What is
the minimum amount of information necessary to constitute life? What is the
smallest constituent of matter? Of energy? What is the furthest extent of the
universe? In mathematics and much of our everyday experience, scale seems
arbitrary. One could construct a house at 1/5 scale, 1:1 scale, and 2:1 scale
without much real difference between them. But once we nudge outside of our
comfort zone, we find that scale is very important in our physical world.
If we
were divide a one kilogram ingot of gold we could cut it down to size and
divide it into tenths, hundredths, thousandths, millionths, billionths, and so
on. But we cannot divide the ingot indefinitely, the smallest unit of gold we
can have is one atom, and there are approximately 3.057x10^24 atoms of gold in
a one kilogram ingot. From that bar we can have no more than around 3 trillion
trillion pieces of gold. Divide it any further, and what you have is no longer
discernibly gold. Given that the properties of matter and energy and various
chemical reactions require certain scales before they can arise (even if those
scales seem imperceptibly small), it seems reasonable to assume that there
exists a fundamental lower limit on size. At some point we must reach a limit
below which there would be no properties at all, since all properties would necessarily
arise from the interactions occurring at such a scale. Viewed this way, a
fractal theory of the universe seems untenable: how can universes be packed
within each other when scale is so important to physical nature? However, it is
still possible that beyond the scale that we can currently observe, past the
fundamental building blocks of our universe, lay the outer reaches of embedded
fractal universes, and these universes have analogous but dissimilar properties
that operate on different magnitudes of scale.
Our Lawless Universe
Space is homogeneous: This assumption states that the equations of physics are independent of the frame of reference of the observer, they are time-space invariant. This assumption is necessarily true, a law that only holds locally is not a law but is at best a subset of a more complicated law or very likely a crude approximation of the true law. Behind the idea of a natural law is the assumption that natural phenomena can be reliably described and predicted by such laws. If on the most fundamental level there is no consistency, that is the laws of nature are variable or in flux and there is no larger pattern or law to this variation, then it is impossible to describe the universe using laws. However, that is inconsistent with experience and all logical thinking. It would take significant philosophical acrobatics if not be outright impossible to imagine a lawless universe that manages to resemble our own in our lifetimes. If a law is true, then it is necessarily applicable to the whole universe, not just our local space. However, it is very likely that a law is incomplete, or an approximation, that only agrees with observation so long as we only test within our neck of the woods. Trying to then apply this law to further reaches then breaks down, since we could not say that it is universally applicable, as we then have yet to discover the universally applicable law that applies beyond the local subset or "special case" law that we have found. Space is necessarily homogeneous, but it is entirely possible to derive physical laws that apply only to our local space. Further such subset laws may be able to account for the vast number of observable phenomena beyond our local space, but should breakdown at scales far beyond our local range. Newton's laws of motion represent a set of approximations that break down far outside of our local space, specifically at speeds approaching those of the speed of light. For a law to offer true insight and provide predictive power, it must extend beyond our local space. However, with new laws and knowledge we expand the horizon of our local space, so that we must probe further at the extremes to gain further insight.
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