Persons without infinities

There is a distant planet, Zeta, whose people are taught math without the concept of infinity. You see, unlike ours, the quantifiers (the existential ∃ and universal ∀) they have in their math are bounded, which means the variable governed by a quantifier ranges over a finite set (according to certain rules, explained below).

On Zeta — as opposed to here on Terra — they write

(∃Z_ix) instead of (∃x),   and

(∀Z_jy) instead of (∀y).

We can think of the Z subindexes ranging over some ordered set ι of indefinite ordinals such that given two of them, i and j with j > i, we can find a k between them: i < k < j. (We can for our purposes assume them to be rational numbers, for example.)

Each Z_i is thought of as some indefinitely large set: a set with an i-zillion elements. Z_ix means that Z_i is large enough to contain x.

Furthermore, in formulas where a Z_j occurs within the scope of a Z_i, j > i. Such a Z_j (also called a domain) is larger than Z_i, but — and this is a key point — Z_j doesn't have to include every element of Z_i.

Removing the Z_is from all their formulas gives our Terran formulas. There is a number of Zetan mathematicians though (the Platonists) who study this variation of math that looks like ours. But this number is small.

(The Zetans think in terms of the indefinitely large, not infinity, and the indefinitely small, not continuity.)

Their Axiom of Zillion (corresponding to our Axiom of Infinity) looks like this:

(∃Z₀x)(∅∈x ∧ (∀Z₁y)(y∈x → (y∪{y})∈x))

This axiom says that there is some indefinitely large set Z₀ which contains a set x that contains the empty set, and from Z₀ a larger indefinitely large set Z₁ can be reached (that is, is in the scope of Z₀) such that for its elements y that are also in the set x, the successor of y — defined as succ(y) = y∪{y} — is in x as well.

Now for Z₀ and Z₁ to be finite, we have to have a case where there is some Z₁n such that succ(n)∈x but where this successor is not in Z₁ itself: n∪{n}∉ Z₁ — this is so x and Z₁ can have some greatest element that breaks the progression to Terran infinity. (This is where it's the case that Z₁ need not — and in this case must not — include all the elements Z₀.)

If the Zs mentioned in the Axiom of Zillion can be superseded by larger Zs then there seems to be a clear sense in which all the numbers in the set x cannot be "all" the numbers (that is, in the Terran way of looking at math). We can get bigger and bigger Zetan numbers, some without an immediate successor or an immediate predecessor. Think of gaps in their numbers, and of very huge numbers out by themselves, all alone. (The Zetans are particularly fond of incredibly huge numbers. And incredibly small ones too, it turns out.)

Going further, in Zetan mathematics, they have their own Cantor's theorem: They also can prove their "integers" — they say there are a bazillion of them — cannot be put into a one-to-one matchup with their "reals" —there's a gazillion of them — which is really bigger). And their own Continuum Hypothesis, and so on ...

. . . . . .       .   ....  . ...        .

It turns out every theorem of Zetan math is a theorem of Terran math, and vice versa — the two maths are isomorphic (or congruent) — due to a theorem by Jan Mycielski^*:

Zetan math   ≅   Terran math

^* This post is based on §5 of uni-duesseldorf.academia.edu/ManuelBremer/Papers/896122/Varieties_of_Finitism and the references therein (e.g., see jstor.org/discover/10.2307/2273942).

The Acts Of The Holy Ghost ~ Scene 9

• Heaven. HAROLD, H.G., JESSE, GABRIEL, and MICHAEL.
•  
• HAROLD
• I can't believe it all worked out like this. We had to intervene and put a stop to this.
• JESSE
• What happened?
• H.G.
• We brought you up here two Earth days ago. I know it probably seems like a dream now. The guards who were taking you to be executed woke up after we knocked them out. Unfortunately, they thought they would be killed instead by Cai and the law teachers for their fuck up so they found this crazy guy they knew about, drugged him, and took him to be crucified. It didn't turn out well for him.
  
  The next evening, we stole the body of that unfortunate guy — don't know his name — from the tomb he was buried and buried him somewhere else. People are running around now thinking you arose from the dead somehow.
• JESSE
• What do I do now?
• H.G.
• I'll take you back for a while so you can make an appearance ...

The Acts Of The Holy Ghost ~ A Play In One Act

Easter for materialists

Materialism is what there is without the holy smoke and mirrors. So what do materialists think about during the long Easter weekend — Maundy Thursday, Good Friday, Black Saturday, Resurrection Sunday, Easter Egg Roll Monday?

The celebrated "events" are adequately covered by Mel Gibson's The Passion of the Christ (2004) — except for the appearance of the Easter Bunny at the Easter Egg Roll. And there's Cecil B. DeMille's The Ten Commandments (1956), which fills in the Passover story of Moses for the important context of the Last Supper of Jesus and his disciples (on Maundy Thursday evening). Of course there's the four Gospels, which nobody reads, each book telling varying versions of the story. A time to catch up, at least, on significant cultural icons of arts and schmaltz.

So what do all these Passover Suppers, Passions of Jesus, and Easter Egg Rolls leave us with? I'm not sure, but the restaurants serving Easter Sunday Brunches should be full.

Bas van Fraassen, philosopher of science

Today is philosopher of science Bas van Fraassen's birthday (72th), so what better reason to read what he's about.

Richard Rorty linked him into his pantheon of anti-essentialists (anti-Platonists^*):

Some day, intellectual historians may remark that the twentieth century was the one in which the philosophy professors began to stop asking bad questions – questions like “What really exists?” “What are the scope and limits of human knowledge?” and “How does language hook up with reality?” These questions assume that philosophy can be done ahistorically. They presuppose the bad idea that inspection of our present practices can give us an understanding of the “structure” of all possible human practices.

“Structure” is just another word for “essence.” The most important movements in twentieth-century philosophy have been anti-essentialist. These movements have mocked the ambitions of their predecessors, positivism and phenomenology, to do what Plato and Aristotle had hoped to do – sift out the changing appearances from the enduringly real, the merely contingent from the truly necessary. Recent examples of this mockery are Jacques Derrida's Margins of Philosophy and Bas van Fraassen's The Empirical Stance. These books stand on the shoulders of Heidegger's Being and Time, Dewey's Reconstruction in Philosophy, and Wittgenstein's Philosophical Investigations. All these anti-essentialist books urge us to fight free of the old Greek distinctions between the apparent and the real, and between the necessary and the contingent.

"Anti-clericalism and atheism", Richard Rorty

Some van Fraassen quotes:

"The success of current scientific theories is no miracle. It is
not even surprising to the scientific (Darwinist) mind. For any
scientific theory is born into a life of fierce competition, a jungle
red in tooth and claw. Only the successful theories survive -
the ones which in fact latched on to the actual regularities in
nature."
quotes.dictionary.com/To_develop_an_empiricist_account_of_science_is

"To develop an empiricist account of science is to depict it as
involving a search for truth only about the empirical world,
about what is actual and observable.... It must involve
throughout a resolute rejection of the demand for an explanation
of the regularities in the observable course of nature, by means
of truths concerning a reality beyond what is actual and
observable, as a demand which plays no role in the scientific
enterprise."
quotes.dictionary.com/The_success_of_current_scientific_theories_is_no

Fans of Stephen Hawking and Leonard Mlodinow's The Grand Design (from two scientists, not philosophers) should be able to relate:

Model-dependent realism is a controversial philosophical approach to scientific inquiry, which accepts that reality can always be interpreted in a number of different ways, and focuses on how well our models of phenomena. It claims that it is meaningless to talk about the "true reality" of the model. The only meaningful thing is the usefulness of the model. The term itself was coined by Stephen Hawking and Leonard Mlodinow in their 2010 book, The Grand Design.
en.wikipedia.org/wiki/Model-dependent_realism

From Richard Rorty's review of van Straassen's The Empirical Stance:

Readers of this book are likely to hope that it will soon be supplemented by one in which van Fraassen tells us more about what sort of projects he has in mind, and about their relevance to the academic discipline of philosophy. He grants that philosophy is “for the most part an academic enterprise, that is, also objectifying” (p. 177), but he does not discuss how that enterprise might be of use to non-objectifying inquirers. He describes his book as “a personal response to philosophy as I found it” (p. xvii), saying that he is an analytic philosopher who views the analytic revolution in philosophy as “subverted by reactionary forces”. But he has not yet made clear what analytic philosophy might look like once those forces have been overcome.

(One can't really split philosophy from politics.)

Finally, some Quine:

As an empiricist I continue to think of the conceptual scheme of science as a tool, ultimately, for predicting future experience in the light of past experience. Physical objects are conceptually imported into the situation as convenient intermediaries -- not by definition in terms of experience, but simply as irreducible posits18b comparable, epistemologically, to the gods of Homer. Let me interject that for my part I do, qua lay physicist, believe in physical objects and not in Homer's gods; and I consider it a scientific error to believe otherwise. But in point of epistemological footing the physical objects and the gods differ only in degree and not in kind. Both sorts of entities enter our conception only as cultural posits. The myth of physical objects is epistemologically superior to most in that it has proved more efficacious than other myths as a device for working a manageable structure into the flux of experience.

Two Dogmas of Empiricism
Willard Van Orman Quine

So we have Wittgenstein, Dewey, Quine, Heidegger, Derrida, Rorty and van Fraassen: a mix of anti-essentialists. Or whatever you want to call them!

Does that put materialism in a box? I don't think so. In Huw Price's (Naturalism Without Mirrors) identification with subject naturalism (vs. object naturalism), philosophers' and scientists' attempts at representations (language, mathematics) are part of the same world as what they are attempting to represent.


Materialism is just what you get without (holy) smoke and mirrors.

^* Plato was one of the first essentialists, believing in the concept of ideal forms, an abstract entity of which individual objects are mere facsimilies. To give an example; the ideal form of a circle is a perfect circle, something that is physically impossible to make manifest, yet the circles that we draw and observe clearly have some idea in common — this idea is the ideal form. Plato believed that these ideas are eternal and vastly superior to their manifestations in the world, and that we understand these manifestations in the material world by comparing and relating them to their respective ideal form. Plato's forms are regarded as patriarchs to essentialist dogma simply because they are a case of what is intrinsic and a-contextual of objects — the abstract properties that makes them what they are. For more on forms, read Plato's parable of the cave.
en.wikipedia.org/wiki/Essentialism

The Acts Of The Holy Ghost ~ Scene 8

• Earth. JEESE tried before HIGH PRIEST CAI and THE COUNCIL. Sends him to the GOV. PYLE for execution.
•  
• CAI
• Is it true that you say you've said you are the one and only son of God? I have a witness.
• JESSE
• Well, I didn't pick my father. Or my dad.
• CAI
• And is it true you teach that the we are no longer under the Laws of Moses?
• JESSE
• Well, ...
• CAI
• Well, I'll have none of that. I vote we turn him over to Governor Pyle to be executed. The Romans are always looking for a Jew or two to crucify. Let's give him JESSE! Who votes 'Aye'?
• 
  [THE COUNCIL (all raise hands). JESSE led away.
  
  GOVERNOR PYLE and JESSE.
  
   H.G., MICHEL, GABRIEL attack guards in hallway
   and take control of JESSE.]

The Acts Of The Holy Ghost ~ A Play In One Act