Posts tagged ‘infinity’

June 9th, 2013

The Philosophy of Science Directory

by Max Andrews

This is a compilation of posts, which focus on the philosophy of science. These posts will cover a broad spectrum within the philosophy of science ranging from multiverse scenarios, scientific theory, epistemology, and metaphysics.

  1. MA Philosophy Thesis: “The Fine-Tuning of Nomic Behavior in Multiverse Scenarios”
  2. Natural Law and Scientific Explanation
  3. Science and Efficient Causation
  4. Which Comes First, Philosophy or Science?
  5. The Postulates of Special Relativity
  6. There’s No Such Thing as Creation Science–There’s Just Science
  7. Time Travel and Bilking Arguments
  8. “It’s Just a Theory”–What’s a Scientific Theory?
  9. Exceptions to a Finite Universe
  10. Teleology in Science
  11. Duhemian Science
  12. The Relationship Between Philosophy and Science
  13. The History of the Multiverse and the Philosophy of Science
  14. Where’s the Line of Demarcation Between Science and Pseudoscience?
  15. Miracles and the Modern Worldview
  16. Mass-Density Link Simpliciter
  17. Scientific Nihilism
  18. Q&A 10: The Problem of Defining Science
  19. Q&A 6: Scientism and Inference to the Best Explanation
  20. The Quantum Universe and the Universal Wave Function
  21. The History and Macro-Ontology of the Many Worlds Interpretation of Quantum Physics
    read more »

May 14th, 2013

Infinity + 1 = ?

by Max Andrews

When we think of infinity we usually think of the usual two categorical distinctions:  a potential infinite and an actual infinite.  A potential infinite suggests that infinity is only an idea or a concept but doesn’t actually exist in the Platonic sense or in the physical sense. In any set, one may always be added.  An actual infinite is the notion that there exists such a set, Platonic or physical, whichis infinite.  A potential infinity may be symbolized by a lemniscate:  ∞.  An actual infinite can be depicted by the aleph-null or aleph-nought:  ℵ0 (The Hebrew letter aleph with a subscript zero).

First, let’s have a brief refresher on set theory. A set is any collection of things or numbers that belong to a well-defined category.  In a set notation, this would be written as {2, 3, 5, 7, 11} being the first five prime numbers, which is a finite set of things.  Let’s simply signify this set as S.  There is a proper subset (SS) of S.  There are members in S that are not in SS, but no member of SS that is not in S.  

December 10th, 2012

Q&A 1: Kalam and The Flying Spaghetti Monster

by Max Andrews

Hey Max,

I guess since I requested the Q&A section, I’ll start it off!

I recently had a conversation with an atheist in which I walked him through the Kalam Cosmological Argument. This inevitably led into a conversation about what criteria a “first cause” must meet. It was difficult for me to explain, and for him to understand how God exists as a necessary being, or out of His own nature.

The atheist resorted to a version of  “Flying Spaghetti Monster” argumentation, in which he said, “How do we know that the first cause wasn’t a giant pink unicorn, or that two universes didn’t just mate and form ours?”. For obvious reasons, his argument is absurd. But what’s the best way to explain the concept of the first cause, and why it couldn’t be a “giant pink unicorn”?

Thanks a lot,

Richie Worrell (USA)

Richie,

I’m always amazed at some of the philosophical lunacy some atheists come up with. The mockery of using phrases like “flying spaghetti monster” or a “giant pink unicorn” weren’t originally developed in response to the kalam. They were developed in response to intelligent design suggesting the designer could be a spaghetti monster. I recall Dawkins using it several times and it has gained popularity in response to the ontological argument.

Nonetheless, let’s accept his flying pasta, pink unicorn, and sexual universes for the sake of discussion. Let’s recap the the kalam argument:

  1. Whatever begins to exist has a cause of its existence.
  2. The universe began to exist.
  3. Therefore, the universe has a cause of its existence.
    read more »

February 8th, 2012

Word of the Week Wednesday: Multiverse

by Max Andrews

The Word of the Week is: Multiverse

Definition: The term to designate the existence of many worlds or universes.  Contrary to just one world, a uni-verse, there are many worlds, a multi-verse.

More about the term: The multiverse is not monolithic but it is modeled after the contemporary understanding of an inflationary model of the beginning of this universe suggesting a plurality of worlds.  Max Tegmark has championed the most prominent versions of the multiverse.[1]  There are four levels of the multiverse.

  1.  Level One:  The level one is, for the most part, more space beyond the observable universe.  So, theoretically, if we were to go to the “edge” of the universe there would be more space.  Having this model as a version of the multiverse may be misleading because there is still only one volume, landscape, or system involved.  A generic prediction of cosmological inflation is an infinite space, which contains Hubble volumes (what we see in our universe) realizing in all conditions—including an identical copy of each of us about 10^10^29 meters away.[2]
    read more »

January 17th, 2012

William Lane Craig’s “J. Howard Sobel on the Kalam Cosmological Argument”–A Review

by Max Andrews

A Review of William Lane Craig’s “J. Howard Sobel on the Kalam Cosmological Argument.” Canadian Journal of Philosophy 36 (2006): 565-584.

William Lane Craig formulates retort to J. Howard Sobel’s objection to kalam as he typically formulates it.[1] Premise 1 seems obviously true—at least, more than its negation.  To suggest that things could just pop into being uncaused out of nothing is to quit doing serious metaphysics and is a premise that Sobel acknowledges to be true.  Sobel’s objection is with 2—that the universe began to exist.  This would then run into an infinite regress, which is philosophically and mathematically untenable.  Because an actually infinite number of things cannot exist, the series of past events must be finite in number and, hence, the temporal series of past, physical events is not without beginning.[2]

December 31st, 2011

An Outline of Tegmark’s Four Levels of the Multiverse

by Max Andrews

Contemporary physics seem to indicate that there are good reasons, theoretically and physically, for an idea that there is a plurality of worlds.  This concept has come to be understood as the multiverse.  The multiverse is not monolithic but it is modeled after the contemporary understanding of an inflationary model of the beginning of this universe.  Max Tegmark has championed the most prominent versions of the multiverse.[1]  Tegmark has made a four-way distinction.

Tegmark’s first version of the multiverse is called the level one multiverse.  The level one is, for the most part, more space beyond the observable universe.  So, theoretically, if we were to go to the “edge” of the universe there would be more space.  Having this model as a version of the multiverse may be misleading because there is still only one volume, landscape, or system involved.  A generic prediction of cosmological inflation is an infinite space, which contains Hubble volumes (what we see in our universe) realizing in all conditions—including an identical copy of each of us about 10^10^29 meters away.[2]

November 22nd, 2011

The Different Versions of Infinity and Cantorian Sets

by Max Andrews

When we think of infinity we usually think of the usual two categorical distinctions:  a potential infinite and an actual infinite.  A potential infinite suggests that infinity is only an idea or a concept but doesn’t actually exist in the Platonic sense or in the physical sense. In any set, one may always be added.  An actual infinite is the notion that there exists such a set, Platonic or physical, which is infinite.  A potential infinity may be symbolized by a lemniscate:  ∞.  An actual infinite can be depicted by the aleph-null or aleph-nought:  ℵ0 (The Hebrew letter aleph with a subscript zero).

First, let’s have a brief refresher on set theory. 

June 14th, 2011

The Metaphysics of the Kalam Cosmological Argument

by Max Andrews

William Lane Craig is perhaps the most well known contemporary proponent of the kalam cosmological argument.  It was during his Ph.D. studies at the University of Birmingham in England where he revitalized this early argument originally developed by the Islamic theologian Al-Ghazali (1058-1111).  There are several cosmological arguments such as Aquinas’ hierarchical causal argument and Leibniz’s principle of sufficient reason, which suggests that everything has a sufficient reason for its existence.  The kalam argument focuses on the impossibility of an infinite set of causes in a temporal manner.

A Few Preliminaries

  • A set is any collection of things or numbers that belong to a well-defined category.  In a set notation, this would be written as {2, 3, 5, 7, 11} being the first five prime numbers, which is a finite set of things.  Let’s simply signify this set as S.
  • There is a proper subset (SS) of S.  There are members in S that are not in SS, but no member of SS that is not in S.
  • The set of first three primes in a proper SS of the above S is {2, 3, 5}.
  • A dense set is a set where there is always room for one more in between another two elements.
  • Where there is an infinite set is with a set of cardinality, or natural numbers, it’s simply called a power set or an infinite set.
  • A series is an ordered set of numbers.  A finite series has a finite fixed number of terms.  An infinite series has an infinite number of terms.  A series with m terms, or the sum of the firs m terms of an infinite series, can be written as Sm or ∑a.
  • An actual infinite set is signified by the Hebrew letter aleph א.
  • potential infinite set, or series, is signified by the lemniscate ∞.

The Argument

  1. Whatever begins to exist has a cause of its existence.
  2. The universe began to exist.
  3. Therefore, the universe has a cause of its existence.

The argument for premise 1 is that anything that begins to exist does so temporally, at some indexical moment of time.  Because there is a difference between moments, an earlier or later than, there must be a cause to the thing which begins to exist, which determines its temporal existence.  Craig offers two arguments for premise 2

2.1 Argument based on the impossibility of an actual infinite

  • 2.11 An actual infinite cannot exist
  • 2.12 An infinite temporal regress of events is an actual infinite
  • 2.13 Therefore, an infinite temporal regress of events cannot exist

2.2 Argument based on the impossibility of the formation of an actual infinite by successive addition

  • 2.21 A collection formed by successive addition cannot be actually infinite
  • 2.22 The temporal series of past events is a collection formed by successive addition
  • 2.23 Therefore, the temporal series of past events cannot be actually infinite

Premise 3 follows necessarily if 1 and 2 are true and valid.  So, what type of cause are we looking at?  Let’s take a look at Aristotle’s causes.

Aristotle’s Four Causes

  1. Material Cause:  Out of what?  This is where the physics come in.  Hawking, for example, agrees with Craig on this, the universe had no material cause. The universe is from nothing.
  2. Efficient Cause:  Through what?  This is the type of cause kalam gets to.  This is why Craig argues that the cause must be a personal agent.  An agent is the only entity that could initiate or cease a series of cause and effect relationships.  This is known as agent causation.
  3. Formal Cause:  What form or essence?
  4. Final Cause:  For what purpose?  This is teleological argument.

Objections

Paul Draper’s “In time or In or with time

Paul Draper responds to P1.  He believes Craig seems to elide over a distinction, and a very important one right at the heart of the debate. There is a difference between:

  1. 1’ Whatever begins to exist in time has a cause of its existence; and,
  2. 1’’ Whatever beings to exist in or with time has a cause of its existence

It is 1’’ that gets back to the main issue: is the universe caused?  The issue here is that time began withthe universe.  It may be true that all things we’ve ever observed that started to exist [in time] had a cause, and eve, that they must have had one.  Still, the unknown (Humean) question is, is 1’’ true on the basis of 1’’? The truth of 1’ seems irrelevant to the inquisitive truth of re: 1’’ Craig’s possible reply? There is still the issue of creatio continuans, that is, God as a cause of contingency’s existence from moment to moment.  Reply? Yes, perhaps; but that isn’t what the kalam argument is supposed to focus on.  The focus has changed from the creation of the universe in the finite past to sustaining the universe at each moment (which is more of Thomas’ argument).  I think this may be one of the stronger objections but the objection seems to interpolate the original premise for what it was.  There’s the difference between concurrent causation in or with time and the beginning of time.

Alvin Plantinga’s division of time…

Plantinga (Warranted Christian Belief, Ch. 1) argues that Craig assumes that each moment of time is of equal duration.  If we divine up time like the following, then there are an actual infinite number of time points in any finite time segment.  Count the events going back in time… 1 second, ½ sec., ¼ sec., etc.  Craig’s reply? Time isn’t like that though, cosmological time is quantal; that is, there is an actual smallest amount of time, which is Planck time (10-43 seconds).  It is at the increment of time can any meaningful physical event take place (this point works in conjunction to Draper’s objection as well).  If we wanted to chase this rabbit trail we would have to then get in to the metaphysical aspects of time and whether or not time is one smooth flow or choppy like a film strip (and at an incredibly choppy rate of one frame per 10-43 second).  This means Plantinga’s example won’t work, and so if each event in the past is taken to be Planck time, then there cannot be an actual infinite number of past events, for reasons previously mentioned.

J. A. Cover’s appeal to omniscience…

Cover’s objection is with 2.11.  Wait a second, but what about God?  Doesn’t God know an actual infinite number of propositions?  Theologians and philosophers have always said that God is “infinite”?  Is his infinity and actual infinity or a potential infinity?  Response?  It assumes a Platonic idea of divine omniscience.  Craig has come to seem less conceptualistic, that is, that the abstract like numbers and certain ideas exist in the mind of God, and has seemed to embrace nominalism.  This nominalism would suggest that propositions do not exist.  Propositions do not exist in a form of a set or a series; rather, propositions are useful fictions.  There’s a distinction Thomas makes in omniscience, there’s a difference between an intuitive knowledge (or non-propositionally) and a discursive knowledge.  I’m not too certain how far Craig take’s any of Thomas’ material but the Thomistic idea of divine approximation may work.  With this idea, God is the ultimate archetype.  If God is simple, but the object of knowledge is not himself, is that not complexity?  So, if God knows a plant, God knows the plant by approximating it to himself (that it exists, that it lives, etc.).  I would venture to say that Craig parts in this area but kalam is still compatible with this concept of God (though Thomas himself would disagree about the nature of time).

Conclusion

The conclusion of the argument ends with a first temporal cause.  Now there are some implications that may be made.  This cause must be personal (nature of agent causation), extremely powerful (observing the effect), timeless (at least explanatorily prior to the beginning of the universe), and changeless (nature of events).  Kalam does not arrive at God, but it ends where it ends… a first temporal cause.