16 June, 2014

Julio Koko Sosa

Yesterday, tío Julio "Koko" Sosa died. His passing came more quickly than I expected.

Julio was a master of the guitar. I was always mesmerized anytime he would play during one of our family get-togethers. The entire family will miss him greatly.

Julio performed, composed, and arranged for hundreds of recordings, concerts, and festivals over the years. He is well remembered in his hometown.  Until very recently, he taught regularly in DC. Some of his students upload their covers of his songs on youtube. You can see him perform at the Kennedy Center below, or in El Salvador in the embedded video below that.

09 July, 2013


Jack at home.
We first met six months ago. He was friendly and cute; inquisitive and demanding. By our third meeting, he had me give him my full attention, playing with toys and petting him continuously. Soon afterward, when I became his housemate, I would feed him twice a day and sometimes share a bed with him. We interacted daily -- him by purring, me by talking. We had become not just housemates, but close friends. Family.

This morning, he woke me up by jumping atop me, and then curling up against my head. I stroked his fur before even opening my eyes. Two hours later, while at work, I received news: Jack had died.
It was a wholly unexpected death. He was at the vet getting his teeth cleaned. He never woke from the anesthesia.

A sudden death like this is especially upsetting. He was only six years old. Far too young. I already feel the lack of him in my life, and I only just got the news an hour ago.

Jack loved attention.
It is harder, of course, for those who knew him longer than I. I was only the latest in an ever growing family who cared for him. Katherine is distraught. Many others have already expressed their deep sympathies. But I'm worried most about Jasper.

Jasper has been by Jack's side for years. They have not been apart for more than 24 hours in years. They have been constantly in each other's presence for 364 days each year since 2008. But Jasper is a cat, and so does not know English. He cannot view the body. I do not know how to explain to him that his adoptive brother has died.

This is a sad day for me. More so for Katherine. But for Jasper, it will be mostly confusing, I think. Which is a sad situation all its own.

Jack in the box.
And to Jack, whose consciousness has now passed, let me say that I am sorry your life has ended. Death is the most terrible of all evils. We must do what we can to combat this foe, and in your name I will take a small step by donating $250 to Vegan Outreach, one of the two most effective charities listed on Effective Animal Activism that help to avert the unnecessary suffering and death of non-human animals like Jack. According to the best research we have, this should help to alleviate between 15 and 25,000 years of nonhuman animal suffering. I trust Peter Hurford's estimate the most, which says that my $250 donation should result in approximately 17,280 total years of suffering averted.

In the meantime, I must comfort Katherine and try to get across what has happened to Jasper. This is not a fun day.

Jack and Jasper wishing me a happy birthday earlier this month.

Goodbye, Jack. You are loved. You are family. <3

02 January, 2013

A New Year

I've never been much of a fan of new years' celebrations. I never put much stock into it as a holiday, and I've never done anything particularly memorable on such a day. Well, other than a y2k prank I did by turning off the power in a house party at midnight on December 31, 1999.

Yet today I am nevertheless looking forward to the new year. Times have been especially hard on me lately, and after my car was towed this morning, it's just been somewhat demoralizing. Life has good times and bad times; at the moment, I merely need to ride out these lows until I can do a bit better in the coming year.

I realize this doesn't make for a particularly interesting blog entry, but it helps me to say such things in a public space like this, so I'm posting it anyway.

06 December, 2012

Really Big Numbers

EDIT: Portions of this article when first published were factually inaccurate. These errors have since been corrected using strikethrough text.

Occasionally, when I'm talking to younger people and they find out that I'm mathematically literate, I'll get a question of the form: "What's the biggest number you can name?"

It's a somewhat difficult question to answer, because it really depends on what counts as naming and what counts as a number. After all, am I allowed to use a function to identify a number? Does the number have to be finite? Must it be useful in some sense? Should I include transfinite numbers? Or do they just want the highest number that has a common english name?

Since there are different answers depending on what the questioner thinks is acceptable, I'll try to list a few of the standard answers I give to these younger questioners here, starting from the smallest and work my way up. If I've missed any natural answers to the question, please comment with your suggestion, so I can add it to my repertoire.

Common English Names

The first few counting numbers have names that we commonly use in English, e.g., ten, hundred, thousand, million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, &c. Some of these get obscure, but they're nevertheless understandable. One of the largest of these in fact has a famous search engine named after it: a googol is equivalent to 10^100. Google's headquarters is also named after an even larger such number; the Googleplex near San Jose, California, is named after the googolplex, which is 10^(googol). In between these two values is the centillion at 10^303, which is about as large as you can get while using the "-illion" suffix. (You can artificially construct larger terms, like the uncentillion, but this is just a matter of inserting additional prefixes.)

Of note here should be Archimedes' system of estimating the number of grains of sand that could fill the universe at 10^63 by using a unit called the myriad, which is equal to 10,000. Archimedes did this by calling a myriad of myriads the first unit, and then taking a myriad myriad of myriad myriad units to create the second unit, and so on for eight times, saying this would be sufficient to fill the universe with grains of sand.

Mathematical Notation

Of course, there are many numbers that are much larger than these, but they generally do not have common english names. Rather, we write them out using scientific notation or by some other mathematical notation. The googolplex above is only 10^(10^100); if you want something bigger, you need only write out another exponent. But even if you repeatedly take exponents of exponents all day long, you won't get anywhere near where you can get if you instead use a more efficient notation system.

Mathematical operations build upon themselves recursively, as any schoolchild learns when they begin to see the relationship between addition and multiplication. Exponentiation is the next logical step after multiplication, but there are further steps one can take. The next method up is called tetration, and it involves repeated exponentiation.

The addition of A+N is when you take the number A and succeed it N times. The multiplication of A*N (or A×N) is when you take A and add it to itself N times. The exponentiation of A^N (or AN) is when you take A and multiple it to itself N times. The tetration of NA is when you take A and exponentiate it by itself N times. I'm sure you can see the pattern here.

NA is A^A^…^A^A where there are N As in the sequence. This results is very large numbers. 10010 is much bigger than 10100, for example. But some notations go far beyond mere tetration.

One such way of notating large numbers is by using Knuth's up-arrow notation. Rather than use superscript, up arrows are used to indicate which type of operation is being used. Exponentiation is written out as A^N (or A↑N) instead of AN. Tetration is written as A^^N (or A⇈N or A↑2N) instead of NA. And the next step in the series follows the same rule: pentation is written as A^^^N (or A↑↑↑N or A↑3N), which is just taking A and tetrating it N times. Similarly, hexation is A^^^^N (or A⇈⇈N or A↑4N), which is taking A and pentating it N times. And so on.

It is perhaps a little difficult to fully understand the scale of these numbers. Even the innocuous looking 3^^^3 is much, much, much bigger than a googolplex:

Add another arrow or two in there and the number becomes so large that trying to just think about it boggles the mind. Usually, when I want to name a very large finite number, I will go with 3^^^^3. It's far bigger than almost anything that we deal with in reality, and so effectively works as a stand in for "extraordinarily large finite number".

However, it's not even close to the biggest that some mathematicians work with.

Graham's number refers to the number of dimensions a hypercube must have in order to satisfy some esoteric conditions in Ramsey theory. The number is so large that the number of up-arrows you'd need to display it is itself amazingly huge. Here's a lower bound of Graham's number: Here's one upper bound of Graham's number that isn't even the lowest known upper bound:

In the above equation, notice that the number of up arrows in the first line is indicated by the number in the line below it. My "extraordinarily large finite number" that I commonly use, 3^^^^3, is merely the number of up arrows used in the sequence directly above it. As you can see, Graham's number is huge.

Fast Growing Functions

And yet, Graham's number is not nearly as big as some. If functions are allowable in our answer, then much bigger numbers can be shown in a small amount of space. Unfortunately, many people seem to think that saying f(x) where the function is defined as X+1 is not the same thing as naming a number. If I say f(3), we all know that what I'm pointing at is 4, but perhaps we might say that I only pointed at it and yet did not name it. However, there is philosophical debate on this issue; after all, what more is there to naming than pointing out which thing we mean? In a very real sense, f(3) can be said to be yet another name for the number 4 here. If you agree with this logic, then we can easily construct much bigger numbers than even Graham's number by merely using functions that grow much faster than even nested up arrow notation can handle.

Take the busy beaver function. It refers to the maximum number of steps performed by turing machines of a certain class, and has been mathematically proven to grow faster than any computable function, though there might non-computable functions that grow even faster. It starts out slow, with Σ(12) being only 3^^^^3, the bottom line of Graham's number. But it grows fast, where Σ(2k) > 3k-23 > A(k-2,k-2) (k≥2), and A() is Ackermann's function: A(m,n)=2↑m-2(n+3)-3.

But if you think the busy beaver function grows quickly, then consider the TREE(x) function. It was created to deal with trees in set theory and grows much faster than most people can appreciate. TREE(3)=A(A(...A(1)...)), where the number of As is A(187196), and A(x) is a version of Ackermann's function: A(x) = 2↑x-1x. Compare to Graham's number, which is just A64(4) — a much smaller number than the established lower bound of TREE(3) at AA(187196)(1). If you want to a compact way of pointing out an absurdly large finite number, consider using TREE(100). The only way I know of to reliably get higher ordinals is to put in bigger numbers inside the function, and god help you if you start iterating the function inside itself.

Take the TREE(x) function. It was created to deal with trees in set theory and grows much faster than most people can appreciate. TREE(3)=A(A(...A(1)...)), where the number of As is A(187196), and A(x) is a version of Ackermann's function: A(x) = 2↑x-1x. Compare to Graham's number, which is just A64(4) — a much smaller number than the established lower bound of TREE(3) at AA(187196)(1). If you want to a compact way of pointing out an absurdly large finite number, consider using TREE(100).

But if you think the TREE() function grows quickly, then consider the busy beaver function. It refers to the maximum number of steps performed by turing machines of a certain class, and has been mathematically proven to grow faster than any computable function, though there might non-computable functions that grow even faster. It starts out slow, with Σ(12) being only 3^^^^3, the bottom line of Graham's number. But it grows fast, where Σ(2k) is

and A() is Ackermann's function: A(m,n)=2↑m-2(n+3)-3. The only way I know of to reliably get higher ordinals is to put in bigger numbers inside the function, and god help you if you start iterating the function inside itself.

Edit: This section was originally inverted, and incorrectly portrayed the TREE() function as faster growing than the Busy Beaver function.

Infinity and Beyond

Of course, all the above numbers are finite, so there is at least one number that beats them all: ∞. Yet the infinity that most people think of when they imagine "the infinite" is merely א0, the first and lowest transfinite cardinal number. These are countably infinite sets, like the set of all positive integers. Yet this comparatively small infinity dwarfs in comparison to א1, the uncountable set which might correspond to the cardinality of real numbers. (This the continuum hypothesis, and it only works if 2^(א0)=א1, which is far from clear.)

But cardinality is not the only kind of infinity; there are also ordinal infinities. The lowest transfinite ordinal number is ω, which corresponds to the order type of the natural numbers. We can then consider the אa where a=ω. This makes אω, which is the lowest uncountable cardinal number that is not the same as the cardinality of the real numbers. the smallest uncountable cardinal provably in ZFC that is not the cardinality of ℜ. (The cardinality of the real numbers might be א1, which is smaller, or אω+1, which is bigger (or many other cardinals).) Don't worry if this is starting to make you dizzy; it does the same to me, too.

However, this is just the low end of the scale. At the high end, things really get deep. Dedekind looked at the system of all ordinal numbers, calling it Ω. Wondering if he could get even higher, he proposed the sequence:

{0, 1, 2, 3, … ω0, ω0+1, …, γ, …}

where each γ in the sequence goes on to describe the order type of all preceding elements. But this turns out to be inconsistent, since if it were consistent, then there would be some number that corresponds to its order, and that number would have to be higher than any of the numbers in the above sequence. But since this is a number, it must be in Ω to begin with, since Ω has all numbers in it. But a number cannot be greater than itself, and so this new proposed set by Dedekind cannot exist. This means that Ω is definitely the absolute infinity of its class.

Does this mean that Ω is the winner here? Can we get any bigger? The answer, once again, is it depends. Woodin's Ultimate L, LΩ, is the ultimate enlargement of Gödel's constructible universe L, which is itself the proper class of all constructible pure sets. Here, L refers to the standard inner model universe of the inner model theory ZFC+V=L, where ZFC is Zermelo-Fraenkel set theory and V is the Von Neumann Universe stationary club (closed unbounded class) of all well-founded pure sets.

That's a mouthful, but does it mean that is LΩ "higher" than Ω? I don't know that it even makes much sense to ask the question, because they're talking about different things. Yet they're both "numbers", so maybe that means they can be compared anyway? To be honest, I'm not entirely sure. When it comes to infinities any bigger than א0, my brain implodes. The previous paragraph, in fact, was written with only a vague understanding of what I've read elsewhere — it's more regurgitation than true understanding on my part.


After all this, I suppose the biggest entity I feel comfortable talking about is Ω, especially when it comes to trying to answer unexpected questions from younger people. Concepts like Hausdorff's absolute continuum cΩ or Woodin's Ultimate LΩ describe situations that may or may not correspond to אω or Ω or whatever. For all I know, everything from אω and up are all equivalently large entities. At some point I may bother trying to understand this stuff more in depth, but until then I think I'll just stick with the large finite numbers and Ω if they want to know how big ∞ can truly get.

EDIT: A few mathematicians expressed concern that some of the mathematical sentences I originally presented in this essay were factually incorrect. These errors have since been corrected, and previous text has been left in using strikethrough text to illustrate the errors I originally ran into as a layperson attempting to write a mathematical article. Thank you to professor Edgar Bering and grad students Bo Waggoner and Charlie Cunningham for finding and correcting my errors.

11 November, 2012

My Philosophical Positions

As promised in a previous entry, I have decided to make my answers to the 2009 PhilPapers Survey public on my blog. These answers were previously made public in a LessWrong poll, but in the interest of making my specific positions easy to read, I've compiled them all here. Note that this post is intended solely to give my position on each of these issues, and does not currently have commentary on why I feel as I do. However, in future entries I intend to zero in on each issue; as I do so, I plan to link to those entries from these position statements in order to more explicitly share why I hold these particular philosophical positions.

My answers are given in bold. The distribution of answers from the 2009 PhilPapers survey is given as a percentage after each answer. Note that percentages for "accept" and "lean toward" are combined here, although more finely grained results are available on the PhilPapers site. A short explanation (thanks to pragmatist & PhilPapers' clarifications for help) of each question is provided for any readers that are not well versed in these issues. Links to future blog posts on why I hold each position will be made available as I write them. Links to relevant SEP articles are provided.

The 2009 PhilPapers Survey

A priori knowledge: yes or no?
  • Accept: yes (71.1%)
  • Lean toward: yes
  • Accept: no (18.4%)
  • Lean toward: no
  • Other (10.5%)
Yes: There exist facts we can know without our knowledge being based on sensory experience.
No: Justification of knowledge requires sensory experience.

Abstract objects: nominalism or Platonism?
  • Accept: nominalism (37.7%)
  • Lean toward: nominalism
  • Accept: Platonism (39.3%)
  • Lean toward: Platonism
  • Other (23.0%)
Abstract objects are objects that do not correspond to any pattern of matter/energy in space-time.
Nominalism: Abstract objects do not exist.
Platonism: Abstract objects exist.

Aesthetic value: objective or subjective?
  • Accept: objective (41.0%)
  • Lean toward: objective
  • Accept: subjective (34.5%)
  • Lean toward: subjective
  • Other (24.5%)

Analytic-synthetic distinction: yes or no?
  • Accept: yes (64.9%)
  • Lean toward: yes
  • Accept: no (27.1%)
  • Lean toward: no
  • Other (8.1%)
Yes: Some sentences are true solely due to the meanings of the words.
No: Every sentence is open to empirical falsification or no sentence is open to falsification.

Epistemic justification: internalism or externalism?
  • Accept: internalism (26.4%)
  • Lean toward: internalism
  • Accept: externalism (42.7%)
  • Lean toward: externalism
  • Other (30.8%)
Externalism: Belief can be justified even when the justification os not consciously available to the subject.
Internalism: Belief is only justified if there is conscious understanding of the justification.

External world: idealism, skepticism, or non-skeptical realism?
  • Accept: idealism (4.3%)
  • Lean toward: idealism
  • Accept: skepticism (4.8%)
  • Lean toward: skepticism
  • Accept: non-skeptical realism (81.6%)
  • Lean toward: non-skeptical realism
  • Other (9.2%)
Idealism: Reality is not mind-independent.
Skepticism: Mind-independent reality exists, but we lack epistemic access to it.
Non-skeptical realism: Mind-independent reality exists, and we have epistemic access to its structure.

Free will: compatibilism, libertarianism, or no free will?
  • Accept: compatibilism (59.1%)
  • Lean toward: compatibilism
  • Accept: libertarianism (13.7%)
  • Lean toward: libertarianism
  • Accept: no free will (12.2%)
  • Lean toward: no free will
  • Other (14.9%)
Compatibilism: We can have free will in a deterministic universe.
Libertarianism: Incompatibilism is true and we have free will.
No free will: Free will does not exist.

God: theism or atheism?
  • Accept: theism (14.6%)
  • Lean toward: theism
  • Accept: atheism (72.8%)
  • Lean toward: atheism
  • Other (12.6%)
Theism: Gods exist.
Atheism: Gods do not exist.

Knowledge: empiricism or rationalism?
  • Accept: empiricism (35.0%)
  • Lean toward: empiricism
  • Accept: rationalism (27.8%)
  • Lean toward: rationalism
  • Other (37.2%)
Empiricism: Only sensory experience gives us new information.
Rationalism: Some information exists that we can arrive at without sensory experience.

Knowledge claimscontextualismrelativism, or invariantism?
  • Accept: contextualism (40.1%)
  • Lean toward: contextualism
  • Accept: relativism (2.9%)
  • Lean toward: relativism
  • Accept: invariantism (31.1%)
  • Lean toward: invariantism
  • Other (25.9%)
Contextualism: The truth of a knowledge claim depends on the context in which it is uttered.
Relativism: Whether a subject possesses knowledge of a proposition is relative to a set of epistemic standards.
Invariantism: The truth of knowledge claims does not depend on context and is not relativized to epistemic standards.

Laws of nature: Humeanism or non-Humeanism?
  • Accept: Humeanism (24.7%)
  • Lean toward: Humeanism
  • Accept: non-Humeanism (57.1%)
  • Lean toward: non-Humeanism
  • Other (18.2%)
Humeanism: The laws of nature are compressed descriptions of salient patterns in the distribution of physical events.
Non-Humeanism: The laws of nature are not mere descriptions, but actually determine the distribution of physical events.

Logic: classical or non-classical?
  • Accept: classical (51.6%)
  • Lean toward: classical
  • Accept: non-classical (15.4%)
  • Lean toward: non-classical
  • Other (33.1%)
Classical: Standard logics, such as Boolean logic or first-order predicate calculus, are best (or correct).
Non-classical: The best logic is not classical (e.g., paraconsistent logic).

Mental content: externalism or internalism?
  • Accept: externalism (51.1%)
  • Lean toward: externalism
  • Accept: internalism (20.0%)
  • Lean toward: internalism
  • Other (28.9%)
Externalism: The representational content of our mental states is dependent upon properties of our external environment.
Internalism: The representational content of our mental states is fixed by our brain state.

Meta-ethics: moral realism or moral anti-realism?
  • Accept: moral realism (56.4%)
  • Lean toward: moral realism
  • Accept: moral anti-realism (27.7%)
  • Lean toward: moral anti-realism
  • Other (15.9%)
Moral realism: Objective moral facts exist.
Moral anti-realism: Objective moral facts do not exist.

Metaphilosophy: naturalism or non-naturalism?
  • Accept: naturalism (49.8%)
  • Lean toward: naturalism
  • Accept: non-naturalism (25.9%)
  • Lean toward: non-naturalism
  • Other (24.3%)
Naturalism: All causes are natural.
Non-naturalism: Supernatural causes exist.

Mind: non-physicalism or physicalism?
  • Accept: non-physicalism (27.1%)
  • Lean toward: non-physicalism
  • Accept: physicalism (56.5%)
  • Lean toward: physicalism
  • Other (16.4%)
Physicalism: A physical duplicate of our world must necessarily also be a mental duplicate.
Non-physicalism: Mental states are not dependent on physical states.

Moral judgment: cognitivism or non-cognitivism?
  • Accept: cognitivism (65.7%)
  • Lean toward: cognitivism
  • Accept: non-cognitivism (17.0%)
  • Lean toward: non-cognitivism
  • Other (17.3%)
Cognitivism: Moral statements have truth conditions.
Non-cognitivism: Moral statements have no truth conditions.

Moral motivation: internalism or externalism?
  • Accept: internalism (34.9%)
  • Lean toward: internalism
  • Accept: externalism (29.8%)
  • Lean toward: externalism
  • Other (35.3%)
Internalism: A necessary connection exists between sincere moral judgment and either justifying reasons or motives.
Externalism: Any connection that exists between moral judgment and motivation is purely contingent.

Newcomb's problem: two boxes or one box?
  • Accept: two boxes (31.4%)
  • Lean toward: two boxes
  • Accept: one box (21.3%)
  • Lean toward: one box
  • Other (47.4%)
Omega appears before you with two boxes and says you may take Box A or take both Box A and Box B. Omega has almost certain predictive power and does not lie. Omega has predicted which you will choose; if Omega predicts you will take just Box A, then Box A will contain $1,000,000. Box B always contains $1,000. How many boxes do you take?

Normative ethics: consequentialism, deontology or virtue ethics?
  • Accept: consequentialism (23.6%)
  • Lean toward: consequentialism
  • Accept: deontology (25.9%)
  • Lean toward: deontology
  • Accept: virtue ethics (18.2%)
  • Lean toward: virtue ethics
  • Other (32.3%)
Consequentialism: The morality of actions depends only on their consequences.
Deontology: There are moral principles that forbid certain actions and encourage other actions purely based on the nature of the action itself, not on its consequences.
Virtue ethics: Ethical theory should not be in the business of evaluating actions, but in the business of evaluating character traits.

Perceptual experience: disjunctivism, qualia theory, representationalism, or sense-datum theory?
  • Accept: disjunctivism (11.0%)
  • Lean toward: disjunctivism
  • Accept: qualia theory (12.2%)
  • Lean toward: qualia theory
  • Accept: representationalism (31.5%)
  • Lean toward: representationalism
  • Accept: sense-datum theory (3.1%)
  • Lean toward: sense-datum theory
  • Other (42.2%)
Disjunctivism: In normal cases, when a person is perceiving something, the object of their perception is a mind-independent object.
Representationalism: Perceptual experience is representational.
Sense-datum theory: The objects of our perception are not mind-independent entities, they are mind-dependent objects called sense-data.
Qualia theory: The phenomenal character of our perceptual experience is non-representational.

Personal identity: biological view, psychological view, or further-fact view?
  • Accept: biological view (16.9%)
  • Lean toward: biological view
  • Accept: psychological view (33.6%)
  • Lean toward: psychological view
  • Accept: further-fact view (12.2%)
  • Lean toward: further-fact view
  • Other (37.3%)
Physical view: The maintenance of personal identity requires bodily continuity.
Psychological view: The maintenance of personal identity requires continuity of psychological states.

Politics: communitarianism, libertarianism, or egalitarianism?
  • Accept: communitarianism (14.3%)
  • Lean toward: communitarianism
  • Accept: libertarianism (9.9%)
  • Lean toward: libertarianism
  • Accept: egalitarianism (34.8%)
  • Lean toward: egalitarianism
  • Other (41.0%)

Proper names: Fregean or Millian?
  • Accept: Fregean (28.7%)
  • Lean toward: Fregean
  • Accept: Millian (34.5%)
  • Lean toward: Millian
  • Other (36.8%)
Fregean: The meaning of a proper name is a way of conceiving of its bearer.
Millian: The meaning of a proper name is its bearer.

Science: scientific anti-realism or scientific realism?
  • Accept: scientific anti-realism (11.6%)
  • Lean toward: scientific anti-realism
  • Accept: scientific realism (75.1%)
  • Lean toward: scientific realism
  • Other (13.3%)
Scientific anti-realism: There are no strong reasons to believe in their theoretical claims about unobservable entities (though epistemic justification of predictions exist).
Scientific realism: There are strong reasons to believe in the theoretical claims about unobservable entities made by our best scientific theories.

Teletransporter (new matter): survival or death?
  • Accept: survival (36.2%)
  • Lean toward: survival
  • Accept: death (31.1%)
  • Lean toward: death
  • Other (32.7%)
You are placed in a machine that will instantaneously disintegrate your body, in the process recording its exact atomic configuration. This information is then beamed to another machine far away, and in that machine new matter is used to construct a body with the same configuration as yours. Would you consider yourself to have survived the process, and teleported from one machine to the other ("survival")? Or do you think you have died, and the duplicate in the far away machine is a different person ("death")?

Time: B-theory or A-theory?
  • Accept: B-theory (26.3%)
  • Lean toward: B-theory
  • Accept: A-theory (15.5%)
  • Lean toward: A-theory
  • Other (58.2%)
B-theory: Specifying the temporal ordering of all events in space-time exhausts all the objective temporal facts about those events.
A-theory: Specifying the temporal ordering of all events in space-time does not exhaust all the objective temporal facts about them.

Trolley problem (five straight ahead, one on side track, turn requires switching): switch or do't switch?
  • Accept: switch (68.2%)
  • Lean toward: switch
  • Accept: don't switch (7.6%)
  • Lean toward: don't switch
  • Other (24.2%)
There is a trolley traveling along a set of tracks. The driver has lost control of the trolley. On the track ahead of the trolley are five people who cannot get off the track in time and will all die if the trolley gets to them. You are standing next to a lever that can switch the track the trolley will take, preventing the deaths of the five people. On the other track is a single person who also cannot get away in time and so will die if you switch the track. Do you refrain from switching the track or do you switch the track?

Truth: correspondence, deflationary, or epistemic?
  • Accept: correspondence (50.8%)
  • Lean toward: correspondence
  • Accept: deflationary (24.8%)
  • Lean toward: deflationary
  • Accept: epistemic (6.9%)
  • Lean toward: epistemic
  • Other (17.5%)
Correspondence: A proposition is true if and only if it bears some sort of congruence relation to a state of affairs that obtains.
Deflationary: Ascribing truth to a proposition amounts to no more than asserting the proposition.
Epistemic: To say that a proposition is true is just to say that it meets a high standard of epistemic warrant, and that we are thereby justified in asserting it.

Zombies: inconceivable, conceivable but not metaphysically possible, or metaphysically possible?
  • Accept: inconceivable (16.0%)
  • Lean toward: inconceivable
  • Accept: conceivable but not metaphysically possible (35.6%)
  • Lean toward: conceivable but not metaphysically possible
  • Accept: metaphysically possible (23.3%)
  • Lean toward: metaphysically possible
  • Other (25.1%)
A zombie is physically identical to a human being but does not possess phenomenal experience. There is nothing it is like to be a zombie.

Inconceivable: We cannot fully conceive of a zombie.
Conceivable but not metaphysically possible: One can arrive at a coherent conception of zombies, but objects that match this conception cannot possibly exist, not even in worlds with different laws of nature than ours.
Metaphysically possible: The existence of zombies is possible.