
Fractal Food: Self-Similarity on the Supermarket Shelf (2005) - ch4ch4
http://www.fourmilab.ch/images/Romanesco/
======
jakub_g
Chou romanesco is probably a regular thing for people living in southern
Europe, but coming from central Europe, it blew my mind the first time I've
seen it in a grocery store :)

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webnrrd2k
If you like this, then you'd almost certainly love the book "The Computational
Beauty of Nature". It explores a lot of similar themes, and is very visual,
too.

[https://mitpress.mit.edu/books/computational-beauty-
nature](https://mitpress.mit.edu/books/computational-beauty-nature)

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justifier
How do those holding opposing views of mathematical realism account for the
presence of such plants and similar computed natural phenomena?

~~~
chestervonwinch
As I understand it, math realism says that the mathematical object (for
example, a class of self-similar sequences) exists whether or not it was
discovered by humans. That the mathematical object happens to model some
physical phenomenon pretty well is independent of the philosophy of
mathematical realism.

Are you arguing against mathematical realism by saying certain mathematical
objects are _only_ discoverable through the observation and modeling of
physical phenomenon? A pure math realist would respond that the object in
question stills exists, whether or not it was discovered.

~~~
ubernostrum
Opposition to mathematical realism also tends to bring in heavy doses of
mathematical+cultural relativism, i.e. believing that if a culture decided 2 +
2 = 5 or that the interior angles of a triangle sum to 360 degrees, then those
statements would be just as true as "2 + 2 = 4" or "the interior angles of a
triangle sum to 180 degrees".

~~~
justifier
but this is more about definitions right?

culture 1 : 2+2=4

culture 2 : 2+2=5

translation between culture 1 and 2:

2==2

4==5

180==360

if you could agree on specific definitions: this is a triangle, this is
interior, this is an angle, and this is a degree; you would be unable to come
to opposing conditions on the sum of interior angles

~~~
ubernostrum
Not really. The relativists aren't saying "oh, your 4 translates to their 5",
and in fact are trying to say that such a statement is impossible to make.
Your attempt at translation assumes the existence of some culture-independent
thing, and tries to say "4" and "5" are different culture-specific symbols for
representing that culture-independent thing, whatever it may be. The
relativist position is that no such culture-independent thing exists.

In other words, the relativist believes that:

* If a culture decides this many objects (represented by dots): ". ." combined with this many other objects: ". .", produce this many objects: ". . . . ."

* Then that decision is just as valid and just as _true_ as our culture's decision that ". ." objects and ". ." objects make ". . . ." objects.

The fact that every culture we know of has adopted the second one instead of
the first is then explained as some sort of massive coincidence, or perhaps
the result of some very old "2 + 2 = 4" culture successfully imposing its
norms on everyone else such that it has persisted as a cultural belief to the
present day.

~~~
justifier
then how would a relativist, as you describe it, justify the forms inherent in
this vegetable.. i mean, what's the 'culture' of vegetables?

~~~
ubernostrum
I suppose on further reflection that it might help to pull back the curtain a
bit and talk about what's really going on here, because this is an ancient
philosophical debate (and I do mean "ancient" in a literal sense -- we're
about to go all the way back to the Greeks here).

"Relativism" is a modern term for one side in one variation of this debate.
The traditional names for the sides would be "realism" and "nominalism". And
to understand the question they're arguing about, I'll ask a question: is a
hot dog a sandwich?

The internet has taught us there are many people who feel strongly that a hot
dog _is_ a sandwich, and many other people who feel just a strongly that a hot
dog is _not_ a sandwich. They argue about how to define the term "sandwich"
\-- does "sandwich-ness" come from the presence of bread? Must there be a
certain number of pieces of bread? Must the bread be in one of a small set of
permitted shapes? What types of things can be in or on the bread?

These people are searching for some thing they can predicate of all
sandwiches. Some property that is indisputably possessed in common by every
sandwich everywhere. What sort of thing is that?

Anyone who's sat through an intro to philosophy course probably knows Plato's
answer. We are, metaphorically, chained to a rock in a cave, facing the back
wall. We know there must be a source of brilliant light (perhaps outside the
mouth of the cave), since although we can't see it directly we see the light
reflected on the cave wall and can tell what direction it's coming from. We
know there are other things out there, too, because there are shadows cast on
the wall as they move back and forth in front of the light. The metaphor, of
course, is that objects in the world are like the shadows cast on the wall of
the cave: distorted representations of true forms which exist somewhere else,
but that we recognize despite the odd angles or movements they make. Thus we
get Plato's theory of forms, which postulates an ethereal-ish realm,
inaccessible to us mortals, in which the ideal forms of things exist. We
recognize disparate things in this world as sandwiches because they all, to
some extent, partake of the Form of Sandwich. Or in cave-metaphor terms, they
all are shadows cast on the wall by the Ur-Sandwich moving about in front of
the light, and though they may look superficially quite different from each
other, we recognize the common origin of them all (Plato had a quasi-religious
explanation for our knowledge of the forms and what they were).

To be a realist is to be committed, on some level, to a theory not dissimilar
from Plato's. We can dress it up in nicer terminology, and cover up some of
the seeming absurdity of there being an ideal Form of Sandwich in some astral
plane somewhere, but ultimately this is what realism says. In some cases it's
used to talk about properties (such as color: "red" may refer to many
different hues, but we use a single umbrella term for all of them -- in what
sense do they all share "redness"?). In mathematical realism, it is used to
talk about (roughly) theorems of mathematical systems. Mathematical realism is
a commitment to the existence of these theorems in some form which is
independent of human minds; the theorem would exist, and be a theorem,
regardless of whether any human ever discovered or proved it.

In many ways this is a very convenient way to talk about mathematics (as well
as to talk about a lot of other things). It allows us to say that "two and two
make four" and "deux et deux font quatre" are in some way the same statement,
though expressed in different words. It gives us an entity which can tie
together many disparate things and serve as the commonality between them all.

But it also seems ludicrous and overcomplex to a lot of people. One of the
earliest critiques of Plato's theory of forms was that they must be infinite
in number (in brief: given any set of objects and a form they all partake of,
it is possible to prove the necessity of another form to tie together the
first form and the objects; from there, yet another can be proven as necessary
to tie together the objects and the first two, and so on to infinity). Willard
Van Orman Quine joked at the proliferation of entities in such systems
(leading to absurdities such as the existence of nonexistence), dubbing it
"Plato's beard" and claiming that it would dull the edge of Occam's razor.

Enter nominalism. Nominalism, simply put, says that there is no Form of
Sandwich, or property of "sandwich-ness", or anything else that all sandwiches
have in common apart from being given the label "sandwich" by people.
Membership in the set of sandwiches -- or any of the other categories realists
invent entities to provide commonality for -- is arbitrary, and we identify
membership in that set not by recognizing some common property possessed
independently of the label, but by rote memorization or heuristic application
of cultural guidelines.

Nominalism, then, would deny that a mathematical theorem has any existence
independent of the minds of humans. It does not sit timelessly, waiting to be
discovered: it is created, it is invented, by humans, and insofar as it
follows rules or has properties, it does so only because humans have
constructed it, the rules, and the properties.

After the ever-increasing complexity of realism, nominalism can seem like a
breath of fresh air, a welcome clearing of the tangled thicket of strange
entities realists sooner or later end up committed to.

But, of course, nominalism comes at a price, and that price turns out to be
heavy. A nominalist cannot admit that, say, your vegetable's physical
appearance and a mathematical formula have anything truly in common other than
a human desire to label them as such. The formula is not a natural thing, to a
nominalist, and does not exist independently of the humans who invented it.
The realist believes the formula would continue to exist, and the vegetable
would continue to follow it, even if all humans suddenly vanished. But the
nominalist _must_ deny this, and say that the vegetable only "follows" the
formula because humans have said it does, and if the humans all vanished there
would no longer be any humans to say things about the vegetable. Similarly,
the nominalist must say that spiral galaxies would no longer be spiral,
because this is a concept created by humans; absent humans to label them, they
would not be spiral in any meaningful sense.

From there, the logical conclusion is more or less absolute cultural
relativism. If the vegetable only follows a formula because we say it does,
what if some other group of people came along and said it didn't? What if they
had an entire belief system, as well-developed and complex as our own, for the
vegetable following some quite different formula? Who would be right in that
case? Since there is no independently-existing "real" formula for the
vegetable to "really" follow, the answer is nobody can say who is right; we
can only say that both sets of beliefs about the vegetable are equally human-
created and equally believed by their respective groups.

This is, of course, incredibly unsatisfying and at odds with how much we've
accomplished by believing that there is real correspondence between
mathematics and the physical world. But to some people it's a better option to
choose over the mess they feel realism will inevitably land them in.

~~~
justifier
How can nominalism be defined for a nominalist?

How does a nominalist account for Godel?

