Archive for ‘Mathematics’

August 31st, 2014

The Likelihood Principle of Confirmation

by Max Andrews

Whenever we are considering two competing hypotheses, an observation counts as evidence in favor of the hypothesis under which the observation has the highest probability.

The Likelihood Principle of Confirmation theory states as follows.  Let h1 and h2 be two be competing hypothesis (in this case the existence of X and ~X, with X being a first cause, fine-tuner, a particle, etc.).  According to the Likelihood Principle, an observation e counts as evidence in favor of hypothesis h1 over h2 if the observation is more probable under h1 than h2.  Thus, e counts in favor of h1 over h2 if P(e|h1) > P(e|h2), where P(e|h1) and P(e|h2) depict a conditional probability of e on h1 and h2, respectively.  The degree to which the evidence counts in favor of one hypothesis over another is proportional to the degree to which e is more probable under h1 than h2: particularly, it is proportional to P(e|h1)/P(e|h2) .  The Likelihood Principle seems to be sound under all interpretations of probability.  This form is concerned with epistemic probability.

November 17th, 2013

What Are the Odds?

by Max Andrews

The odds against an event is a ratio of the probability that the event will fail to occur (failure) to the probability that the event will occur (success).

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Most of the time you hear about “odds”, they are referring to the odds against. 

November 15th, 2013

Theoretical Probability

by Max Andrews

Below is a brief introduction to theoretical probability. Probability is something that is used very, very frequently in philosophy and science. I would encourage further research and familiarity with it than just this.

Definition:  If each outcome of an experiment has the same chance of occurring as any other outcome, they are said to be equally likely outcomes.

Let E be an event having equally likely outcomes, then the probability of E may be calculated with the following formula:

Important facts:

  1. The probability of an event that cannot occur (impossible event) is 0.  p(ø) = 0
  2. The probability of an event that must occur is 1. p(S) = 1
  3. Every probability is a number between 0 and 1, inclusive.  0 < p(E) < 1
  4. The sum of the probabilities of all possible outcomes of an experiment is 1.
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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 26th, 2013

A Missing Argument from March’s Debate

by Max Andrews

In preparation for my debate this past March I went through my opponents blog to find relevant arguments he may use. (He didn’t use any I had expected. Instead, he just did the scatter-gun approach by putting little, ineffective arguments out there hoping they’d stick instead of presenting a few robust, substantive arguments.) I found this argument on the impossibility of God from omniscience and omnipotence. I expected this because these are the types of arguments atheists should be using. In order to demonstrate a universal negative one must demonstrate that the referent is impossible or logically incoherent; that is, since contradictions are the only things that cannot obtain. To much misfortune, the argument was not presented. If it had been presented the debate would have been much more substantive. (To watch the debate for yourself, watch it here. Don’t take my word for it.)

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.  

August 27th, 2012

The Language of God

by Max Andrews

In our experience, intentions get actualized any number of ways[1]: A sculptor by chiseling at stone, musicians by writing notes, engineers by drawing up blueprints. In general, all actualizations of intentions can be realized in language. Precise enough sets of instructions in a natural language can tell the sculptor how to form the statue, musician how to record the notes, and engineer how to draw up blueprints.

Why should an act of speech be God’s mode of creation? Language is the universal medium for actualizing intentions. The language that proceeds from God’s mouth in the act of creation is the divine Logos (Jn. 1.1-5). In the act of creation God the Father speaks the divine Logos in the power of the Holy Spirit. The divine Logos is not just language in the ordinary sense (utterances that convey information), but the very ground and possibility of language. Words need power to accomplish their end and God’s Word has that power (Is. 55.11).

Given that we are made in God’s image, the Trinitarian structure of creation is reflected in human speech.

“The word [goes] out of the mouth of God in such a manner that it likewise ‘[goes] out of the mouth’ of men; for God does not speak openly from heaven, but employs men as his instruments, that by their agency he may make known his will.”[2]

June 26th, 2012

The Multiverse, Fine-Tuning, and Nomic Probabilities

by Max Andrews

Whenever probability is being considered there must be some type of relevant or total background information (usually depicted as k).  The immediate objection when applying a probability rule or calculus to the fine-tuning of the universe in a multiverse scenario would be to say that this is universe is not an appropriate random sampling.  In other words, if we know of [at least] only one universe with these values the random sample size is precisely 1; thus, no random sample can be used to assess the probability of certain values of physics in the argument.  In statistics a random sample drawn must have the same chance of being sampled as all the other samples.  Since we know of only one universe we do not know what the range of values for the constants and physics could be.  Additionally, since we don’t know how narrow or broad these ranges could be there’s no way of drawing out any probability based argument for fine-tuning.  However, we can know what other universes would be like if the values were different.  If our natural laws have counterfactuals that are in any way incoherent then this is an appropriate sampling.  Also, to make this objection and advocate that we just so happen to live in a life permitting universe in the multiverse then this objection cannot be made since the claim that we happen to life in a life-permitting one amongst countless others suggest we can know what the other samplings are.

May 9th, 2012

Word of the Week Wednesday: Mathematical Invariance

by Max Andrews

Word of the Week: Mathematical Invariance

Definition:  In Einstein’s use of the word, mathematical invariance established a genuine ontology in which the subject grips with objective structures and intrinsic intelligibility of the universe.

More about the word:  Throughout Einstein’s work, the mechanistic universe proved unsatisfactory.  This was made evident after the discovery of the electromagnetic field and the failure of Newtonian physics to account for it in mechanistic concepts.  Then came the discovery of four-dimensional geometry and with it the realization that the geometrical structures of Newtonian physics could not be detached from changes in space and time with which field theory operated. 

April 18th, 2012

Information Theory

by Max Andrews

Information theory is the branch of probability theory that deals with uncertainty, accuracy, and information content in the transmission of messages. It can be applied to any system of communication (electric signals, fiber optic pulses, speech, etc.). Random signals, known as noise, are often added to a message during the transmission process, altering the signal received from that sent.  Information theory is used to work out the probability that a particular signal received is the same as the signal sent.  In transmitting a sequence of numbers, their sum might also be transmitted so that the receiver will know that there is an error when the sum does not correspond to the rest of the message.  The sum itself gives no extra information, simply a confirmation.  The statistics of choosing a message out of all possible messages (letters like the alphabet or binary digits for example) determines the amount of information contained in it.  Information is measured in bits (binary digits). If one out of two possible signals are sent then the information content is one bit.  A choice of one out of four possible signals contains more information although the signal itself might be the same.

For more information see John Daintith and John Clark’s The Facts on File Dictionary of Mathematics (New York: Market Book House, 1999), 97.