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Philosophy of Mathematics/Scope

This is an introductory meta chapter on the topic. It answers the basic questions about the scope of the course and gives information for deciding on taking the course or not.

Why studying Philosophy of Mathematics?
This is not about either studying PoM or doing (i.e. studying) nothing at all, the question is not 'to study or not to study'. In fact it's about 'this to study or that to study'.
 * PoM is better than taking another Maths course, since
 * just doing sth without questioning it is like a hamster in a running wheel that never asks what's outside
 * the valuation of mathematical results itself is not done mathematically
 * communications with nonmathematicians from time to time is unavoidable ( such is life ;o)
 * PoM is better than taking another Philosophy of xyz course, since
 * mathematics is THE apriorical science, so here one learns more about the a priori then anywhere else
 * the concept of 'theory' is an elementary one in epistemology, mathematics is where it is defined
 * when talking about mathematics many people appear unconfident, here is clarified what one can and can't expect from it

What does a Philosophy of Mathematics contain?
When doing mathematics 'the usual way' one can easily find that it's not a purely formal science at it's rock bottom. So it can be thought of a mathematical foundation of mathematics. This is usually referred to as metamathematics.

Also one can question the formal method of mathematics at all from a cognitive point of view: is it really better to write confusing lines of symbols then e.g. writing a poem about the stuff? Moreover a mathematician is not alone. There is a whole mathematical community out there and although the results are formal, doing mathematics is highly creative and valuating these results as well as deciding about directions of research is primarily a social problem.

Similarly when thinking about the application mathematics in science, engineering etc. one can take the same two approaches: modelling the situation mathematically and examining the cognitive and community aspects. The application of mathematics can and can't be seen as part of the subject of philosophy of mathematics. Here it is included, but when reasons should appear to source it out to another course this could be done easily.

Exclusions: What it does NOT contain:
 * an introduction to logic or model theory
 * an introduction to epistemology
 * an introduction to cognitive science (should it?)
 * an introduction to social science or game theory (should it?)
 * any training of mathematical abilities or capabilities
 * any issues of non-formal linguistics
 * aesthetics of mathematics (as a standalone subject), since this is truly art and not only a subchapter of philosophy. Although it might appear as part of some philosophical considerations.

How is this course structured?
Problem: Philosophers make statements in epistemology and theory of science. Scientists ignore them mainly or even be annoyed about them (or do you know any formal scientist who loves T.S.Kuhn?). But actually Scientists are interested in the philosophical background of their job as well as Philosophers are interested in the justification of their Theories.

Approach: Structuring the whole subject from the perspective of a questioning Mathematician. Such a course structure may be build up along the following questions, expanding the underlying Model of the 'Mathematician's job' step by step.

So the course will be beneficial for Scientists AND Philosophers as well as it enables both sides also to participate in EDITING the content - leveraging the cooperative spirit of Wikiversity!

title: Philosophy of Mathematics/Scope


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Philosophy of Mathematics/Structure

The subject is primarly structured from the perspective of the reader, i.e. other perspectives like the ones of philosophical schools, people or history are secondary.

Since the main group adressed are people that are doing Maths or interesting in the work of people doing Maths the subject is seen from the perspective of a Mathematician that is questioning what he is doing.

The further structuring is done mainly by analysing the elements of a mathematical result and expanding the notion of the environment in which the mathematical work takes place.

Structuring by the elements of the mathematical result
This questioning takes place step-by-step from 'how to reasonably exploit certain degrees of freedom the Mathematician has' to 'questioning the fundamentals required for the Mathematicians work'.

Hereby the steps are structured along the elements of a mathematical result, starting with the Proof that is being delivered, the Sentence that is chosen to be proven and the underlying Theory and finally the concept of Proof used above as well as the Language of Mathematics itself.

So 'questioning' means deciding on how to use the remaining degrees of freedom like 'I have different ways to do my Proof - is principally one better than the other?' or getting to the bottom of things like 'my Language has some restrictions in expression - what is the cost of choosing a more expressive one?'

Thus we get the following structure:
 * Questioning the Instance of Proof
 * Questioning the Sentence
 * Questioning the Theory
 * Questioning the Concept of Proof
 * Questioning the Language

Structuring by notions of the working environment
Metamathematical results can be obtained without considering the circumstances under which the mathematical results are found.

Above this the nature of PhilM is to go beyond these purely (meta-)mathematical results, i.e. asking questions like '.' or '.'. So it expands the pure mathematical view on how results are achieved with further aspects.

These aspects are achieved by expanding the notion of the Mathematician who is doing the job. First by presuming computational costs for performing Proofs etc, then by taking human aspects like cognition and understanding into account and finally by considering the Scientific Community as a whole.

This results in questions like 'Where do Definitions make sense?' and 'How does information access influent the development of Theories?' that bring in different cognitive, social and efficiency perspectives etc.

Thus we get the following structuring:

 * Metamathematics & Computability
 * Human-Cognition & -Thinking
 * Scientific Communities and their Principals

Joining result-based and enviromental structuring
The above structuring approaches are compatible. They are joined in the following way:

 * Questioning the Instance of Proof
 * Metamathematics & Computability
 * Human-Cognition & -Thinking
 * Scientific Communities and their Principals
 * Questioning the Sentence
 * Metamathematics & Computability
 * Questioning the Language
 * Metamathematics & Computability
 * Human-Cognition & -Thinking
 * Scientific Communities and their Principals
 * Scientific Communities and their Principals

Although the focus of the single areas may vary. While in sections on judgement of mathematical work (using degrees of freedom) human and social issues are more in the centre of interest the focus in questioning the mathematical fundamentals is likely more on mathematical & compuntational issues.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Philosophy of Mathematics/Structure


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Probability and statistics

This curriculum reflects a hybrid between the typical undergraduate and graduate programs in Statistics. It aspires to provide a strong foundation in both the applied and theoretical branches of Statistics. Generally an "undergraduate statistics program" is functionally a math major with an emphasis in some statistical topics. (Rarely will an undergraduate student have the desire or foresight to focus on the field of Statistics quite this much.)

That's okay! Mentioned in this curriculum is the idea of a "statistics minor" which might be a stats emphasis on a math degree or perhaps someone in the applied physical sciences (physics, chemistry, biology, geology, or even psychology) wants to have a strong foundation in experimental design to supplement a research-oriented career. In these cases the student would want to tailor her curriculum with classes up through the fourth semester.

If the student wishes, however, to pursue a real professional career in Statistics, or is considering graduate school, the fifth semester and on will provide an excellent preparation. If anyone actually mastered this entire curriculum, he or she would be on par with any modern graduate student. A full-fledged thesis is expected, and the student will be expected to prepare well in advance starting in the sixth semester so the thesis does not fall under that hurried, last minute curse. Additionally the student will be expected to write a shorter summary paper for submission to two academic journals.

Semester 0: Preparations and Review
Semester 0 is conceptually a collection of preparatory pre-classes intended to give the student a good working foundation before he or she starts the study of Statistics in earnest. These would probably qualify as "half-credit" classes at most. Some students already possessing a strong background in mathematics may opt to test out of these classes and proceed directly to the first semester.

 * Fundamentals of Probability, Statistics, Experiments and Data This discusses the history and nature of "What are Probability and Statistics?" The nature of data (types of data—discrete versus continuous, categorical, etc.) is discussed, as well as topics of problems in data gathering, unintended outside forces getting confounded with the data, etc.
 * Combinatorics This class doesn't require any "fancy high-level math" but it is computationally intensive. Lots of problems will exercise the student's brain, developing that muscle that is expected to understand and visualize issues in probability in later semesters. "Pump it up and feel the burn!"
 * Pre-Statistics Topics in Math, Introduction to Computing with R This is both a review of all the math that will be expected in the first semester (don't worry, it's pre-calculus at best) and an introduction to using R, the Open Source statistical package and programming language that is used in most academic institutions on the graduate level!

Semester 1: Introducing Probability
The first semester provides an introduction to Probability in its true mathematical context. This class attempts to introduce strong foundations while avoiding too much mathematical rigor. (The Probability 2 and Statistics 2 classes will pick up with material that assumes a strong understanding of Calculus.)

 * Introduction to Probability Probability defines the concepts of probability spaces, random variables, distributions, density and distribution functions. Independent and IID (independent identically distributed) random variables are also discussed. It ends with the introduction of expected values, attempting to provide an introduction that is accessible to a student who has not quite yet had a course in calculus.
 * Research Methods and Library Science In today's age so many students don't know how to use a library. The student will be introduced to some of the primary statistical journals and will learn how to conduct research, including modern "interlibrary loan" systems and other means for accessing materials when we don't have a major research library nearby! This class is really intended for the serious student who plans to complete the full major program and attempts to prepare her for the research that will be necessary for thesis work. Fact: you can't just look it all up on the Internet!

Semester 2: Introducing Statistics and Linear Models
The second semester is a continuation under the same pace as the previous semester, with an attempt to establish some foundations and familiarity with statistical and mathematical notation, while avoiding the rigor that requires calculus. (That will come soon enough!) The second half of Regression and Linear Models with attempt to reinforce and restate some of the theory from Introduction to Statistical Analysis in the context of lineal algebra.

 * Introduction to Statistical Analysis Statistical Analysis reviews some fundamental summary statistics and then begins to relate sample statistics with their parallel components in probability. (Sample mean to probability mean, sample variance to variance, etc.) Concepts of confidence intervals and hypothesis testing are introduced with simple examples of each.
 * Regression and Linear Models Theory of regression is presented in the context of Linear Models with a strong emphasis for example. Optional modules that give a solid proficiency in the R computer package are encouraged. Topics from the previous semester's introductory Probability and Statistics courses will be reiterated in the context of Linear Models.

Semester 3: Completing the Basics
These courses complement each other, focusing on a different aspect of the field of Statistics. Whereas Statistics for Experimenters is very "applied" and data-oriented for the experimental or "laboratory" statistician, the time series course will appeal to the pure mathematician and theorist. These classes can be taken concurrently.

 * Statistics for Experimenters This is mostly a review course, discussing all previous material in the context of real-world scientific research. Addition topics will include: best practices for using statistics in "thesis work" in any of the other disciplines, effective writing and presentation in papers and journals, application of the R package, best practices in data gathering. A preview of Sampling and Design of Experiments will hopefully encourage serious researchers to also take those Semester 4 classes.
 * Stochastic Processes and Intro to Time Series Not for the weak of heart, a Statistics Major will now be expected to understand material presented with heavy mathematical notation. Topics in probability are extended into the time-dimension. Concepts of stationarity, ARMA and ARIMA and introduced. The semester ends with an introduction to the kalman filter.

Semester 4: Applied Statistics Topics for Researchers
This semester is very heavy in the "applied statistics" arena. A student pursuing a minor in Statistics would conclude with one of these classes as his or her final class. (One of these classes will be a more natural fit depending on what the specific major program is.)

 * Design of Experiments Class uses the foundations built in Statistics 103: Regression and Linear Models to explore experimental design. This is an example-heavy class with a large amount of work done in R with pre-designed data sets.
 * Sampling All main sampling designs are explored, from Simple Random Sampling to Stratified and Importance Sampling.
 * Categorical Methods and GLIM This class is most useful to students in the social sciences, psychology and pre-med. Contingency tables, Logistical Regression and GLIM will be introduced. This class is also very computer-intensive with sample analyses being performed using R.

Semester 5: Advanced Probability
These classes are not considered for a minor study in Statistics. Probability 2 is a required course for a Statistics major and approaches the level of a first-semester graduate course. From this point all students are expected to have a solid grasp of Calculus.

 * Probability 2 "The rest of probability," this class looks at cumulative distribution functions, moment generating functions, calculation of expectation, conditional probability and random vectors.
 * Computational Statistics Computer modeling, monte-carlo design, kernel density estimation, and other advanced topics.

Semester 6: Advanced Statistics and Thesis Preparation
More of the same: Statistical Analysis 2 has plenty of mathematical rigor. The Advanced Topics are purely considered electives. The student at this point is expected to prepare for a final project, consulting with an advisor and fulling documenting the experimental or sampling design.

 * Statistical Analysis 2 If this class doesn't kill you, nothing will. All the theory leading up to this point will be brought to bear on topics of minimal sufficient statistics, maximum likelihood, confidence intervals and hypothesis testing.
 * Advanced Topics: Clinical Trials An introduction to the application of statistics to clinical trials in the pharmaceutical industry. Includes discussions of Phase I-IV clinical trials. Some independent reading of literature (ie. statistical journals) will be expected.
 * Advanced Topics: Consulting Lab This class is geared to prepare a statistical consultant to help outside experimenters seeking help. Case studies will examine common problems and misconceptions, communication issues, and differences between being brought in early (for planning and design) and being brought in late (a.k.a. "help bring meaning to my worthless data") and everything in between.
 * Thesis Design and Preparation An idea, design, and acceptable levels of background research must be presented to the advisor. Bibliographies of all "background research" materials must be prepared in advance. A detailed calendar must be presented and checked off by the advisor.

Semester 7: Advanced Time Series and Thesis Work
Time Series 2 returns to the time-oriented data with the application of all the Probability and Statistics that has been acquired so far. This semester is generally where the student is supposed to perform the thesis work that had been planned out in the previous semester, including any mid-experiment analysis.

 * Time Series 2 More math analysis and proofs of theorems. Applied examples and outside readings will show how the theory can be applied and developed for various "real world" problems.
 * Advanced Topics: Bayesian Statistics Examination of Bayesian theory and its controversies. The class will discuss instances where a Bayesian approach has the most advantages.
 * Advanced Topics: Spacial Analysis Theories and testing for regularity and clustering, etc.
 * Thesis Work The outline, introduction and background material and most of the bibliography must be completed before the end of the semester. A draft of the journal paper must be in a similar state of preparedness.

Semester 8: Multivariate and Thesis Defense
The student will defend a thesis during this semester, based on the (guided) planning and execution over the previous two semesters.

 * Multivariate Statistics Everything we've done in Statistics 2 is taken to the n-dimension with an extension from random variables to random vectors. Techniques for analyzing eigenvalues and eigenvectors, transforming into orthogonal axes, etc.
 * Mathematical Statistical Theory A purely optional class extending "Probability and Statistics 2" to the next step. The proofs of advanced concepts like the Weak and Strong Laws of Large Numbers and introduction of Measure Theory are approached. This class can really only aspire to set the foundation for a real class in this arguably doctoral level material.
 * Thesis Defense Student will defend the thesis by mid-semester, perform revisions, and will submit a condensed review of work to at least two industry journals. (They just have to be submitted, but accepted.)

title: Probability and statistics


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Mathematics software

School of Mathematics

The links on this page and the software itself have not been validated or verified to be free of risk. Unless otherwise noted, they have been added by anonymous editors. If any registered editor with a substantial contribution history is willing to validate and certify these programs, please sign the listing as "validated." Otherwise this page exists for research purposes. I suggest googling a name before loading it, to find reviews, unless you very much know what you are doing.

To-do: Validate these programs as safe. Eliminate dead links. To-do: Review these programs, comparing them. --Abd (discuss • contribs) 13:32, 4 April 2014 (UTC)

Free mathematics software (freeware)
The below are open source tools and mostly available on GNU/Linux, and can run on other platforms too:

 * Flox Calc free, powerful script based offline graphing program for windows. Includes special functions to model relativistic travel scenarios.
 * GeoGebra - open source software for dynamic geometry, algebra and calculus for middle/high school and college. Exports interactive web pages.
 * Kig A GUI based geometry editor. Includes some analytical functions and conic sections. Exports to several formats including SVG.
 * Newt - free graphing program for windows
 * Maxima - open source symbolic maths program, like Mathematica and Maple
 * SymPy - open source computer algebra software, written in Python.
 * Axiom - another open source symbolic maths program
 * r- open source version of S-plus, a statistics program
 * TeXmacs - WYSIWYG LaTeX
 * Calc - an interactive RPN calculator package for numerical and symbolic calculations in Emacs
 * Calcpad - free programmable engineering calculator for cloud and desktop
 * gnuplot is a free graphing program for Linux, Windows and Mac
 * GraphCalc - A free, open source 2D/3D graphing application. (Windows)
 * Graph - Graph is an open source application used to draw mathematical graphs in a coordinate system (Windows)
 * GNU Octave - Matlab clone for Linux, Windows and Mac
 * Python numarray, powerful numerical calculations in a powerful programming language
 * scilab - Matlab clone
 * DataMelt - Java clone of Matlab. Also includes scripting languages (Python, Groovy etc) on Java for math and data analysis
 * Qalculate - Multipurpose and very powerful desktop calculator
 * SAGE - is a free open-source computer algebra system that includes programs for research and teaching in algebra, geometry, number theory, and cryptography. SAGE is mainly written in Python and provides interfaces to other mathematics packages.
 * SciPy - SciPy is an open source library of scientific tools for Python
 * Formulator Tarsia - is a free mathematical editor designed for teachers of mathematics creating the activities in a form of jigsaws, rectangular cards or dominos for later use in a class.
 * Armadillo - Fast C++ matrix library with optional interfaces to LAPACK and ATLAS
 * Pari/GP - number theory computations
 * Microsoft Mathematics - very useful math app for Windows
 * Algosim - a fairly advanced numerical mathematical system with graphics and SVG support for Windows

Non-free software

 * Mathematica - Advanced commercial math program. (Linux, Macintosh, Windows)
 * Maple - Mature commercial math program. (Linux, Macintosh, Windows)
 * Maple Studio Free online worksheet for plotting, based on Maple. (Browser with Java-Plugin like Mozilla, Internet Explorer, Safari)
 * MuPad - Another advanced math program with free versions available. (Linux, Macintosh, Windows)
 * Magma - Powerful package for solving computational problems in algebra.
 * Matlab
 * MathCAD
 * MatDeck - Advanced mathematical software with programming at affordable cost
 * SOWISO - Online learning and testing environment for STEM. Teachers can use their interactive courseware and also create their own materials. It is available at no cost to students through software licensing. (Linux, Macintosh, Windows)

title: Mathematics software


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Bayesian statistics

Resampling vs. Bayesian Computation
Typically, the question one attempts to answer using statistics is that there is a relationship between two variables. To demonstrate that there is a relationship the experimenter must show that when one variable changes the second variable changes and that the amount of change is more than would be likely from mere chance alone.

There are two ways to figure the probability of an event. The first is to do a mathematical calculation to determine how often the event can happen. The second is to observe how often the event happens by counting the number of times the event could happen and also counting the number of times the event actually does happen.

The use of a mathematical calculation is when a person can say that the chance of the event rolling a one on a six sided die is one in six. The probability is figured by figuring the number of ways the event can happen and divide that number by the total number of possible outcomes. Another example is in a well shuffled deck of cards, what is the probability of the event of drawing a three. The answer is four in fifty two since there are four cards numbered three and there are a total of fifty two cards in a deck. The chance of the event of drawing a card in the suite of diamonds is thirteen in fifty two (there are thirteen cards of each of the four suites). The chance the event of drawing the three of diamonds is one in fifty two.

Sometimes, the size of the total event space, the number of different possible events, is not known. In that case, you will need to observe the event system and count the number of times the event actually happens versus the number of times it could happen but doesn't.

For instance, a warranty for a coffee maker is a probability statement. The manufacturer calculates that the probability the coffee maker will stop working before the warranty period ends is low. The way such a warranty is calculated involves testing the coffee maker to calculate how long the typical coffee maker continues to function. Then the manufacturer uses this calculation to specify a warranty period for the device. The actual calculation of the coffee maker's life span is made by testing coffee makers and the parts that make up a coffee maker and then using probability to calculate the warranty period.

Experiments, Outcomes and Events
The easiest way to think of probability is in terms of experiments and their potential outcomes. Many examples can be drawn from everyday experience: On the drive home from work, you can encounter a flat tire, or have an uneventful drive; the outcome of an election can include either a win by candidate A, B, or C, or a runoff.

Definition: The entire collection of possible outcomes from an experiment is termed the sample space, indicated as $$\Omega$$ (Omega)

The simplest (albeit uninteresting) example would be an experiment with only one possible outcome, say $$A$$. From elementary set theory, we can express the sample space as follows:

$$\Omega = \{ A \} $$

A more interesting example is the result of rolling a six sided die. The sample space for this experiment is:

$$\Omega = \{ 1,2,3,4,5,6 \}$$

We may be interested in events in an experiment.

Definition: An event is some subset of outcomes from the sample space

In the die example, events of interest might include a) the outcome is an even number b) the outcome is less than three

These events can be expressed in terms of the possible outcomes from the experiment: a) : $$ \{2,4,6\} $$ b) : $$ \{ 1,2 \}$$

We can borrow definitions from set theory to express events in terms of outcomes. Here is a refresher of some terminology, and some new terms that will be important later: $$ \cup $$ represents the Union of two events $$ \cap $$ represents the Intersection of two events $$\{\cdots\}^{c}$$ represents the complement of an event. For instance, "the outcome is an even number" is the complement of "the outcome is an odd number" in the dice example. $$ A \backslash B $$ represents difference, that is, $$A$$ but not $$B$$. For example, we may be interested in the event of drawing the queen of spades from a deck of cards. This can be expressed as the event of drawing a queen, but not drawing a queen of hearts, diamonds or clubs. $$\varnothing$$ or $$\{\}$$ represent an impossible event $$\Omega$$ represents a certain event $$A$$ and $$B$$ are called disjoint events if $$A\cap B = \varnothing$$

Probability
Now that we know what events are, we should think a bit about a way to express the likelihood of an event occurring. The classical definition of probability comes from the following. If we can perform our experiment over and over in a way that is repeatable, we can count the number of times that the experiment gives rise to event $$A$$. We also keep track of the number of times that we perform the same experiment. If we repeat the experiment a large enough number of times, we can express the probability of event $$A$$ as follows: $$P(A) = \frac{N_{A}}{N} $$ where $$N_{A}$$ is the number of times event $$A$$ occurred, and $$N$$ is the number of times the experiment was repeated. Therefore the equation can be read as "the probability of event $$A$$ equals the number of times event $$A$$ occurs divided by the number of times the experiment was repeated (or the number of times event $$A$$ could have occurred)." As $$N$$ approaches infinity, the fraction above approaches the true probability of the event $$A$$. The value of $$P(A)$$ is clearly between 0 and 1. If our event is the certain event $$\Omega$$, then for each time we perform the experiment, the event $$\Omega$$ is observed; $$N_{\Omega} = N$$ and $$P(\Omega)=1$$. If our event is the impossible event $$\varnothing$$, we know $$N_{\varnothing}=0$$ and $$ P(\varnothing) = 0$$.

If $$A$$ and $$B$$ are disjoint events, then whenever event $$A$$ is observed, then it is impossible for event $$B$$ to be observed simultaneously. Therefore the number of times events $$A$$ union $$B$$ occurs are equal to the number of times event $$A$$ occurred plus the number of times $$B$$ occurs. This can be expressed as: $$N(A\cup B) = N(A) + N(B)$$ Given our definition of probability, we can arrive at the following: $$P(A\cup B) = P(A) + P(B)$$

At this point it's worth remembering that not all events are disjoint events. For events that are not disjoint, we end up with the following probability definition. $$ P(A\cup B) = P(A) + P(B) - P(A\cap B)$$ How can we see this from example? Well, let's consider drawing from a deck of cards. I'll define two events: "drawing a Queen", and "drawing a Spade". It is immediately clear that these are not disjoint events, because you can draw a queen that is also a spade. There are four queens in the deck, so if we perform the experiment of drawing a card, putting it back in the deck and shuffling (what statisticians refer to as sampling with replacement, we will end up with a probability of $$\frac{1}{13}$$ for a queen draw. By the same argument, we obtain a probability for drawing a spade as $$\frac{1}{4}$$.  The expression $$P(A\cup B)$$ here can be translated as "the chance of drawing a queen or a spade".  If we incorrectly assume that for this case $$P(A\cup B) = P(A) + P(B)$$, we can simply add our probabilities together for "the chance of drawing a queen or a spade" as $$\frac{1}{13}+\frac{1}{4}$$.  If we were to gather some data experimentally, we would find that our results would differ from the prediction -- the probability observed would be slightly less than $$\frac{1}{13}+\frac{1}{4}$$.  Why?  Because we're counting the queen of spades twice in our expression, once as a spade, and again as a queen. We need to count it only once, as it can only be drawn with probability of $$\frac{1}{52}$$. Still confused?

Proof: If $$A$$ and $$B$$ are not disjoint, we have to avoid the double counting problem by exactly specifying their union. $$A \cup B = A \cup (B \backslash A) $$ so $$P(A \cup B) = P(A \cup (B \backslash A)) $$ $$A$$ and $$B \backslash A$$ are disjoint sets. We can then use the definition of disjoint events from above to express our desired result: $$P(A \cup B) = P(A) + P(B \backslash A)$$ We also know that $$P(B \backslash A) = P(B) - P(B\cap A)$$ so $$ P(A \cup B) = P(A) + P(B) - P(B\cap A)$$ Whew! Our first proof. I hope that wasn't too dry.

Conditional Probability
Many events are conditional on the occurrence of other events. Sometimes this coupling is weak. One event may become more or less probable depending on our knowledge that another event has occurred. For instance, the probability that your friends and relatives will call asking for money is likely to be higher if you win the lottery. In my case, I don't think this probability would change.

Let's get formal for a second and remember our original definition of probability. $$P(A) = \frac{N_{A}}{N}$$ Consider an additional event $$B$$, and a situation where we are only interested in the probability of the occurrence of $$A$$ when $$B$$ occurs. A way at this probability is to perform a set of experiments (trials) and only record our results when the event $$B$$ occurs. In other words $$ \frac{N_{A\cap B}}{N_{B}} $$ We can divide through on top and bottom by $$N$$ the total number of trials to get $$P(A\cap B)/P(B)$$. We define this as 'conditional probability': $$P(A|B) = \frac{P(A\cap B)}{P(B)}$$ which when spoken, takes the sound "probability of $$A$$ given $$B$$."

Bayes' Law

 * On Wikipedia: Bayes' theorem

An important theorem in statistics and the cornerstone of Bayesian statistics is Bayes' Law (also known as Bayes' theorem), which states that

$$ P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$, It is easy to prove. We start with identical expressions for $$P(A\cap B)$$. We know that: $$ P(A\cap B) = P(B\cap A) $$, $$ P(A\cap B) = P(A|B)P(B)$$, and $$ P(B\cap A) = P(B|A)P(A)$$. Since $$ P(A\cap B) = P(B\cap A) $$, $$ P(A|B)P(B) = P(B|A) P(A)$$.

A simple rearrangement of above line gives us Bayes' Law.

Independence
Two events $$A$$ and $$B$$ are called independent if the occurrence of one has absolutely no effect on the probability of the occurrence of the other. Mathematically, this is expressed as: $$P(A\cap B) = P(A)P(B)$$.

Note that this implies that $$ P(A|B) = P(A) $$ That is, adding the information about $$B$$ is irrelevant towards determining the likelihood of $$A$$.

Random Variables
It's usually possible to represent the outcome of experiments in terms of integers or real numbers. For instance, in the case of conducting a poll, it becomes a little cumbersome to present the outcomes of each individual respondant. Let's say we poll ten people for their voting preferences (Republican - R, or Democrat - D) in two different electorial districts. Our results might look like this: $$\{RRRDRRDRRR\}$$ and $$\{DDDDDRDDDD\}$$ But we're probably only interested in the overall breakdown in voting preference for each district. If we assign an integer value to each outcome, say 0 for Democrat and 1 for Republican, we can obtain a concise summary of voting preference by district simply by adding the results together.

Discrete and Continuous Random Variables
There are two important subclasses of random variables: discrete random variable (DRV) and continuous random variable (CRV). Discrete random variables take only countably many values. It means that we can list the set of all possible values that a discrete random variable can take, or in other words, the number of possible values in the set that the variable can take is finite. If the possible values that a DRV X can take are a0,a1,a2,...an, the probability that X takes each is p0=P(X=a0), p1=P(X=a1), p2=P(X=a2),...pn=P(X=an). All these probabilites are greater than or equal zero.

For continuous random variables, we cannot list all possible values that a continuous variable can take because the number of values it can take is extremely large. It means that there is no use to calculate the probability of each value separately because the probability that the variable takes a particular value is extremely small and can be considered zero P(X=x)=0).

title: Bayesian statistics


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Statistics/Introduction

Examples

 * 46% of people polled enjoy vanilla, while 54% prefer chocolate (+/-4% margin of error).
 * A school's graduation rate has increased by 2%.
 * A couple has 4 boys, and they are pregnant again: what is their chance of having another boy?
 * 88% of people questioned feel that it is humane to put stray animals to sleep.

These are basic examples of statistics we see every day, but do we really understand what they mean? With the study of statistics, these 'facts' that we hear every day can hopefully become a little more clear.

Introduction
Statistics is permeated by probability. An understanding of basic probability is critical for the understanding of the basic mathematical underpinning of statistics. Strictly speaking the word 'statistics' means one or more measures describing the characteristics of a population. We use the term here in a more idiomatic sense to mean everything to do with sampling and the establishment of population measures.

Most statistical procedures use probability to make a statement about the relationship between the independent variables and the dependent variables. Typically, the question one attempts to answer using statistics is that there is a relationship between two variables. To demonstrate that there is a relationship the experimenter must show that when one variable changes the second variable changes and that the amount of change is more than would be likely from mere chance alone.

There are two ways to figure the probability of an event. The first is to do a mathematical calculation to determine how often the event can happen. The second is to observe how often the event happens by counting the number of times the event could happen and also counting the number of times the event actually does happen.

The use of a mathematical calculation is when a person can say that the chance of the event of rolling a one on a six sided dice is one in six. The probability is figured by figuring the number of ways the event can happen and divide that number by the total number of possible outcomes. Another example is in a well shuffled deck of cards, what is the probability of the event of drawing a three. The answer is four in fifty two since there are four cards numbered three and there are a total of fifty two cards in a deck. The chance of the event of drawing a card in the suite of diamonds is thirteen in fifty two (there are thirteen cards of each of the four suites). The chance of the event of drawing the three of diamonds is one in fifty two.

Sometimes, the size of the total event space, the number of different possible events, is not known. In that case, you will need to observe the event system and count the number of times the event actually happens versus the number of times it could happen but doesn't.

For instance, a warranty for a coffee maker is a probability statement. The manufacturer calculates that the probability the coffee maker will stop working before the warranty period ends is low. The way such a warranty is calculated involves testing the coffee maker to calculate how long the typical coffee maker continues to function. Then the manufacturer uses this calculation to specify a warranty period for the device. The actual calculation of the coffee maker's life span is made by testing coffee makers and the parts that make up a coffee maker and then using probability to calculate the warranty period.

Experiments, Outcomes and Events
The easiest way to think of probability is in terms of experiments and their potential outcomes. Many examples can be drawn from everyday experience: On the drive home from work, you can encounter a flat tire, or have an uneventful drive; the outcome of an election can include either a win by candidate A, B, or C, or a runoff.

Definition: The entire collection of possible outcomes from an experiment is termed the sample space, indicated as $$\Omega$$ (Omega)

The simplest (albeit uninteresting) example would be an experiment with only one possible outcome, say $$A$$. From elementary set theory, we can express the sample space as follows:

$$\Omega = \{ A \} $$

A more interesting example is the result of rolling a six sided dice. The sample space for this experiment is:

$$\Omega = \{ 1,2,3,4,5,6 \}$$

We may be interested in events in an experiment.

Definition: An event is some subset of outcomes from the sample space

In the dice example, events of interest might include a) the outcome is an even number b) the outcome is less than three

These events can be expressed in terms of the possible outcomes from the experiment: a) : $$ \{2,4,6\} $$ b) : $$ \{ 1,2 \}$$

We can borrow definitions from set theory to express events in terms of outcomes. Here is a refresher of some terminology, and some new terms that will be important later: $$ \cup $$ represents the Union of two events $$ \cap $$ represents the Intersection of two events $$\{\cdots\}^{c}$$ represents the complement of an event. For instance, "the outcome is an even number" is the complement of "the outcome is an odd number" in the dice example. $$ A \backslash B $$ represents difference, that is, $$A$$ but not $$B$$. For example, we may be interested in the event of drawing the queen of spades from a deck of cards. This can be expressed as the event of drawing a queen, but not drawing a queen of hearts, diamonds or clubs. $$\varnothing$$ or $$\{\}$$ represent an impossible event $$\Omega$$ represents a certain event $$A$$ and $$B$$ are called disjoint events if $$A\cap B = \varnothing$$

Probability
Now that we know what events are, we should think a bit about a way to express the likelihood of an event occurring. The classical definition of probability comes from the following. If we can perform our experiment over and over in a way that is repeatable, we can count the number of times that the experiment gives rise to event $$A$$. We also keep track of the number of times that we perform the same experiment. If we repeat the experiment a large enough number of times, we can express the probability of event $$A$$ as follows:
 * $$P(A) = \frac{N_{A}}{N} $$

where $$N_{A}$$ is the number of times event $$A$$ occurred, and $$N$$ is the number of times the experiment was repeated. Therefore the equation can be read as "the probability of event $$A$$ equals the number of times event $$A$$ occurs divided by the number of times the experiment was repeated (or the number of times event $$A$$ could have occurred)." As $$N$$ approaches infinity, the fraction above approaches the true probability of the event $$A$$. The value of $$P(A)$$ is clearly between 0 and 1. If our event is the certain event $$\Omega$$, then for each time we perform the experiment, the event $$\Omega$$ is observed; $$N_{\Omega} = N$$ and $$P(\Omega)=1$$. If our event is the impossible event $$\varnothing$$, we know $$N_{\varnothing}=0$$ and $$ P(\varnothing) = 0$$.

If $$A$$ and $$B$$ are disjoint events, then whenever event $$A$$ is observed, then it is impossible for event $$B$$ to be observed simultaneously. Therefore the number of times events $$A$$ union $$B$$ occurs are equal to the number of times event $$A$$ occurred plus the number of times $$B$$ occurs. This can be expressed as:
 * $$N(A\cup B) = N(A) + N(B)$$

Given our definition of probability, we can arrive at the following:
 * $$P(A\cup B) = P(A) + P(B)$$

At this point it's worth remembering that not all events are disjoint events. For events that are not disjoint, we end up with the following probability definition.
 * $$ P(A\cup B) = P(A) + P(B) - P(A\cap B)$$

How can we see this from example? Well, let's consider drawing from a deck of cards. I'll define two events: "drawing a Queen", and "drawing a Spade". It is immediately clear that these are not disjoint events, because you can draw a queen that is also a spade. There are four queens in the deck, so if we perform the experiment of drawing a card, putting it back in the deck and shuffling (what statisticians refer to as sampling with replacement, we will end up with a probability of $$\tfrac{1}{13}$$ for a queen draw. By the same argument, we obtain a probability for drawing a spade as $$\tfrac{1}{4}$$.  The expression $$P(A\cup B)$$ here can be translated as "the chance of drawing a queen or a spade".  If we incorrectly assume that for this case $$P(A\cup B) = P(A) + P(B)$$, we can simply add our probabilities together for "the chance of drawing a queen or a spade" as $$\tfrac{1}{13}+\tfrac{1}{4}$$.  If we were to gather some data experimentally, we would find that our results would differ from the prediction -- the probability observed would be slightly less than $$\tfrac{1}{13}+\tfrac{1}{4}$$.  Why?  Because we're counting the queen of spades twice in our expression, once as a spade, and again as a queen. We need to count it only once, as it can only be drawn with probability of $$\tfrac{1}{52}$$. Still confused?

Proof: If $$A$$ and $$B$$ are not disjoint, we have to avoid the double counting problem by exactly specifying their union.
 * $$A \cup B = A \cup (B \backslash A) $$

so
 * $$P(A \cup B) = P(A \cup (B \backslash A)) $$

A and B \ A are disjoint sets. We can then use the definition of disjoint events from above to express our desired result:
 * $$P(A \cup B) = P(A) + P(B \backslash A)$$

We also know that
 * $$P(B \backslash A) = P(B) - P(B\cap A)$$

so
 * $$ P(A \cup B) = P(A) + P(B) - P(B\cap A)$$

Whew! Our first proof. I hope that wasn't too dry.

Conditional Probability
Many events are conditional on the occurrence of other events. Sometimes this coupling is weak. One event may become more or less probable depending on our knowledge that another event has occurred. For instance, the probability that your friends and relatives will call asking for money is likely to be higher if you win the lottery. In my case, I don't think this probability would change.

Let's get formal for a second and remember our original definition of probability. $$P(A) = \frac{N_{A}}{N}$$ Consider an additional event $$B$$, and a situation where we are only interested in the probability of the occurrence of $$A$$ when $$B$$ occurs. A way at this probability is to perform a set of experiments (trials) and only record our results when the event $$B$$ occurs. In other words $$ \frac{N_{A\cap B}}{N_{B}} $$ We can divide through on top and bottom by $$N$$ the total number of trials to get $$P(A\cap B)/P(B)$$. We define this as 'conditional probability': $$P(A|B) = \frac{P(A\cap B)}{P(B)}$$ which when spoken, takes the sound "probability of $$A$$ given $$B$$."

/A confusing problem/

Bayes' Law
An important theorem in statistics is Bayes' Law, which states that if we segment the probability space $$ \Omega $$ into disjoint $$B_1,...,B_k $$, then $$ P(B_i|A) = \frac{P(A|B_i)P(B_i)}{\sum_{n=1}^kP(A|B_n)P(B_n)}.$$ As a proof, first note that since $$B_1,...,B_k$$ are disjoint then $$ P(A) = \sum_{n=1}^k P(A|B_n)P(B_n).$$ The theorem follows then by substituting into the conditional probability inequality $$ P(B_i|A) = \frac{P(A\cap B_i)}{P(A)}.$$ For a more detailed explanation, see http://en.wikipedia.org/wiki/Bayes'_theorem

Independence
Two events $$A$$ and $$B$$ are called independent if the occurrence of one has absolutely no effect on the probability of the occurrence of the other. Mathematically, this is expressed as: $$P(A\cap B) = P(A)P(B)$$.

Random Variables
It's usually possible to represent the outcome of experiments in terms of integers or real numbers. For instance, in the case of conducting a poll, it becomes a little cumbersome to present the outcomes of each individual respondant. Let's say we poll ten people for their voting preferences (Republican - R, or Democrat - D) in two different electorial districts. Our results might look like this: $$\{RRRDRRDRRR\}$$ and $$\{DDDDDRDDDD\}$$ But we're probably only interested in the overall breakdown in voting preference for each district. If we assign an integer value to each outcome, say 0 for Democrat and 1 for Republican, we can obtain a concise summary of voting preference by district simply by adding the results together.

Discrete and Continuous Random Variables
There are two important subclasses of random variables: discrete random variable (DRV) and continuous random variable (CRV). Discrete random variables take only countably many values. It means that we can list the set of all possible values that a discrete random variable can take, or in other words, the number of possible values in the set that the variable can take is finite. If the possible values that a DRV X can take are a0,a1,a2,...an, the probability that X takes each is p0=P(X=a0), p1=P(X=a1), p2=P(X=a2),...pn=P(X=an). All these probabilites are greater than or equal zero.

For continuous random variables, we cannot list all possible values that a continuous variable can take because the number of values it can take is extremely large. It means that there is no use in calculating the probability of each value separately because the probability that the variable takes a particular value is extremely small and can be considered zero: P(X=x)=0.

Distribution Functions
show that P(A-B)=P(A)-P(B)

title: Statistics/Introduction


text:
School talk:Physics

Name change
I propose that this school should be named "Physics and Astronomy" since the two fields are essentially inseparable, and that is the link from the "browse" page. Ryanfrei, 16:20, 6 September 2007

An Interesting Link
http://www.westga.edu/~jhasbun/osp/osp.htm This link has some java applets demonstrating various features of physics numerically. It might be useful if the link were placed in proper places around the physics, quantum chemistry and advanced engineering material. The author calls his stuff "Open Source Physics" so he might be interested allowing the incorporation of specific links, parameters and/or applets into interactive lesson plans. Mirwin 05:00, 30 October 2006 (UTC)

title: School talk:Physics


text:
Wikiversity talk:Browse/Old

Descriptions
Could we please bring in a quick description possibly from wikipedia as to what each of the schools cover to make sure people are using the right schools. Or at least clarify what each school includes to insure that various projects and learning material are appropriately categorised. --michael_DesignNZ 23:59, 20 August 2006 (UTC)

Engineering and Technology,

Humanities ,

Interdisciplinary Studies,

Life Sciences ,

Physical Sciences ,

Professions,

Social Sciences ,

Linguistics

Organization
Some time down the line, we should make this look a wee bit more professional. You know, with topic-specific logos and such. Of course, we should really focus on the content first and not the aesthetics. I'm not saying....I'm just saying. --The Winged Self 04:15, 21 August 2006 (UTC)

?
sohail qadeer abbasi

Portals versus Faculties
Doesn't it make more sense to call the Portals "Faculties"? I mean, that's how they are named in a real University...--Dnjkirk 06:09, 7 October 2006 (UTC)
 * Some of the major portals are analogous to faculties in universities. But there are also other portals on wikiversity.--Hillgentleman|User talk 09:30, 1 December 2006 (UTC)

 * "I object to the use of the term "faculty" at Wikiversity. Many English-speaking people will naturally assume that if there is a "Faculty" then there must be individual faculty members who are certified and appointed by an educational institution. Many people use a definition for "faculty" that is like this one: "The group of people who teach the courses at a college or university. The faculty consists of lecturers, instructors, assistant professors, associate professors, and professors." Wikiversity does not have appointed faculty members, so it is misleading to say that we have "Faculties". There is nothing wrong with calling Wikiversity portals, "portals" as is done at Wikipedia." (source) --JWSchmidt 21:34, 16 December 2006 (UTC)
 * More English speakers will naturally assume that if there is a school then there must be individual teachers who are certified and appointed by an educational authority. --Hillgentleman|Talk 22:56, 16 December 2006 (UTC)

 * The words "Faculty of" should be removed simply because this phrase is repeated so frequently, it is hard to read. Let's keep things simple! · Robert Elliott 21:04, 16 December 2006 (UTC)

 * One problem - there is a portal:Physical Sciences and a School:Physics and Astronomy, a portal:Mathematics and a School:Mathematics. There may be other examples.  We want the structure to be clear.--Hillgentleman|Talk 21:14, 16 December 2006 (UTC)
 * So what is the problem? Portals are user-friendly guides that quickly direct browsers to content pages. Schools are content-development projects where Wikiversity editors collaborate to develop learning resources for a subject area. --JWSchmidt 21:34, 16 December 2006 (UTC)
 * The problem is that, if you remove the faculty, or if you wish portal: from the header, and leave only mathematics, scholars would be confused. Even more confusion will arise if you find two different links Mathematics and Mathematics on the same page.  (Or Physical Sciences and Physics and Astronomy.--Hillgentleman|Talk 22:31, 16 December 2006 (UTC)
 * Portals are user-friendly guides that quickly direct browsers to content pages. <--- Do you mean that browsers can avoid the schools altogether?--Hillgentleman|Talk 22:40, 16 December 2006 (UTC)

 * Portals are not used correctly
 * Using Portals as a "user-friendly guides that quickly direct browsers to content pages" is a great idea. However, currently, it is not clear what is a portal and what is not a portal.

 * On the main page of Wikiversity, people are told to "Browse our subject areas and schools: Engineering and Technology · Fine Arts · Humanities · Interdisciplinary Studies · Life Sciences · Physical Sciences · Mathematics · Social Sciences · Practical Arts and Sciences. These are all portals.

 * But when you click on "Browse", you get a different set of major portals

 * Also, when you have a portal called Mathematics and a school called Mathematics, people get confused. When you follow the two paths, you do not arrive at the same information. Instead, you find that there are two partially different set of departments of mathematics.  In this case, the portal does not quickly direct browsers to content pages but rather, the portal goes off in a different direction from the school.  (The problem is the worse in Art and Design.)

 * The idea behind portals is great but that is not what portals are used for currently. This needs to be fixed. Robert Elliott 07:33, 17 December 2006 (UTC)

Templatiziation
If we templatized the sections it might be useful. --Remi0o 22:21, 14 December 2006 (UTC)

 * Suggestion: Go to the portal for Mathematics and a school called Mathematics and create a template so that the "Departments of Mathematics" lists will be the same on both pages.

 * Similarly, go to the portal for Mathematics and a school called Computer science and create a template so that the "Departments of Computers Science" lists will be the same on both pages. Robert Elliott 07:55, 17 December 2006 (UTC)

Medicine Icon
Currently the icon for the medicine section is an animation of a human brain. I think this is a very poor choice. For one, it should not be animated. The rest of the icons are not animated, and the icon should be at least somewhat similar to the other icons, and should also lend itself more to being printed. Plus the animation is very choppy (perhaps just my computer) and is not aesthetically pleasing. Secondly, it a picture of a brain, and a brain is already being used in the icon for psychology (which makes sense). Medice is a hugely broad field, I'm sure there are many many more approprate icons that could be found. For example, perhaps an x-ray of a human skeleton. 142.68.160.138 20:46, 24 December 2006 (UTC)

Unstable browse concept page
Please go here if you would like to help develop a more efficient and user friendly version of the browse page. Perhaps the icons could be removed and the information could still be presented in an interesting way. --Remi 04:46, 21 March 2007 (UTC)

Christianity
Could you please add Christianity Studies under Major Schools, the link is: http://en.wikiversity.org/wiki/Christianity Thanks

title: Wikiversity talk:Browse/Old


text:
Fundamentals of probability and statistics

Why statistics are used
Statistics provide a way to describe the behavior of a person or thing or a collection of persons or things. For instance, sport statistics are used to describe the sports performance of players of the sport. Knowing that one baseball player is a .300 hitter while another player is batting .210 can help a baseball coach know which player needs additional help to improve their batting average.

Statistics provide a way for a person to make decisions when the person is uncertain about the things or quantities involved in a decision. For instance, statistics are used in the insurance industry to make decisions about the insurability of people or things as well as to determine the rate to charge for the insurance. Statistics are used in determining the length of product warranties and to track how good sports players are. The use of statistics are good for making decisions involving the stock market and which lottery tickets to buy.

History of Statistics
Statistics were, in a way, studied by the Ancient Greeks. However, statistics began in earnest during the 1600s.

History of Probability
If we consider that man has evolved through the ages, that is Stone Age, Bronze Age and Iron Age, then we could have a basis of discussion.

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. Probability may actually have a history not defined. Looking at the definition of probability;the likely occurance of an event (define as you find appropriate), it seems to have existed before man walked the earth. The decisions that are made in today's life are all based on the facts of their being likely to occur. Similarly, decisions that were made in stone age, Bronze Age or Iron Age were also based on the facts of occurrence,and the likelihood of their success or failure. In the early ages, there were no generalised educated definitions for probability

This means that every decision, has an element of probability in it. This only implies that from the time man was in the primitive stage to start using tools, probability was playing a vital role. Because there were chances that the tools and methods that were going to be used, could or could not work. Just like dressing up does not necessarily guarantee warmth as a function of clothing, some people dress up for fashion's sake not for either covering up or warmth.

Animals also use probability for their survival. They go hunting expecting to catch prey, or find a meal for themselves. But, there is a possibility that they might or might not catch their prey_ Their success and failure is measured by their instincts to survive or fight of which they do not have a guarantee of success. Taking an example of lions, when they catch prey they have a high success rate that they will have it all; but a school of hyenas due to their large numbers (one hyena will fail) will always scare the lions off their meal.

Mordern day mathematicians, however, have studied many systems and have come up with the formulae that would help one make constructive decisions about certain occurrences. Take an example of lotto tickets, every set of six numbers drawn have a chance of one in more than 12 million chances to win the jackpot; the lotto gurus suggest that one who at least plays 72 sets will reduces the base the more sets played, the higher the chances of winning. The stock exchange decisions and Investment decision are made all basing on the fact that using certain modules there is a higher probability of their success.

For The educated and modern definition and history, please view:http://en.wikipedia.org/wiki/Probability_theory#History

From Ssemondo

Variables and Measurements
There are two basic concepts for a person studying statistics to know and understand: what is a variable and what is a measurement.

A variable represents something which can change. For instance, if we are wanting to determine the batting average of a baseball player, two variables would be (1) number of hits and (2) number of times at bat. To calculate the batting average for a baseball player you must observe the desired behavior, ability to hit the ball successfully, and take measurements of the behavior. In this case the ability to hit the ball is computed by taking measurements of two variables, the number of times the baseball player is batting and the number of times when the baseball player is able to hit the ball successfully.

Different players will have different measurements or counts for these two variables. Some players will hit more often than other players. Some players may miss games due to illness or injury so have fewer times at bat.

Other examples of variables are:
 * number of meals served in a restaurant
 * price of houses in a neighborhood
 * miles driven per gallon of gas consumed
 * number of students in a classroom
 * type of pie served in a restaurant
 * high and low temperature for the day

Notice that of this list of variable examples, the types of measurements are different. Some such as number of meals served in a restaurant are counts, the meals are counted. Some such as price of houses in a neighborhood are financial figures or amounts, the amount of money someone paid for a particular house. Some, such as type of pie served in a restaurant, are types meaning that they are not really numbers at all but are classifications of objects (apple pie, chocolate pie, banana cream pie).

There are four different kinds of measurements. Each of these different kinds have different uses in statistical procedures. Some types of statistical procedures are used only with certain kinds of measurements.

The four kinds of measurements [w:Level of measurement] are:
 * Nominal measurement. The numbers are actually placeholders for names, can be replaced by names, and usually are.  Nominal measurements can be compared to see if they are equal or unequal, the same or different.  There is no other kind of comparison that can be made with nominal measurements.  This type of measurement can be remembered as "nominal means named".
 * Ordinal measurement. The numbers are like nominal measurements with another added feature, ordinal measurements have rank or order.  Ordinal measurements can be compared to see if they are equal or unequal.  They can also be compared to see if one is less than or greater than the other.  However, two ordinal measurements can not be added to each other or subtracted from each other.  Ordinal measurements do not have standard intervals.  This type of measurement can be remembered as "ordinal means ordered nominal".
 * Interval measurement. The numbers are like ordinal measurement with another added feature, interval measurements are separated by the same interval.  This means that interval measurements are based on a standard scale making arithmetic such as addition and subtraction possible.  Interval measurements can be compared to see if they are equal or unequal.  They can also be compared to see if one is less than or greater than the other.  In a comparison, not only can you say one is greater than the other but that you can say how many units greater.  This type of measurement can be remembered as "interval means ordinals with intervals".
 * Ratio measurement. The numbers are like interval measurements with another added feature, ratio measurements have ratios between the possible measurement values.  This means that ratio measurements are based on a standard scale which has a fixed zero point in which the zero point means zero amount.

To better explain the differences, lets take a look at examples of each of these different levels of measurements. In some of these cases we will also convert one level of measurement to another to help you understand the differences between the levels of measurements.

Examples of nominal measurements are:
 * type of pie served in a restaurant. The pies served in the restaurant have different names such as chocolate pie, banana cream pie, apple pie (remember nominal means named)
 * religious preference of students in a class. Each student has a particular religious preference such as Catholic, Baptist, Muslim, Hindu (remember nominal means named)

Examples of ordinal measurements are:
 * finish position in a horse race. Each horse finishes in a sequence, first place, second place, third place (remember ordinal means ordered nominal)
 * taste comparison of drinks. Flavor of drinks are compared and ranked in order of preference, first choice, second choice, third choice (remember ordinal means ordered nominal)

Examples of Interval measurements are:
 * temperature in degrees Celsius. Because the Celsius temperature scale has an arbitrary zero point, the freezing point of water, you can compare 5 degrees Celsius to 10 degrees Celsius to see which is greater.

Examples of ratio measurements are:
 * gallons of water used in flushing toilets. The scale for gallons of water has a non-arbitrary and specific zero point.
 * number of slices of pizza eaten. The scale for number of slices of pizza eaten has a non-arbitrary and specific zero point.

Central Tendency and Descriptive and Inferential Statistics
There are two basic kinds of statistics. The first kind is descriptive statistics which are used to describe a sample or a population from which the sample is drawn. The second kind is inferential statistics which are used to infer from a sample characteristics of the population from which the sample is drawn.

Central tendency is the tendency or direction of a sample or a population. The central tendency is usually expressed as the mean though there are other statistics used to express central tendency. For instance a student's Grade Point Average (GPA) is a measure of central tendency, in this case the tendency of the student's ability to be a good student. Descriptive statistics are the basic computations which we all know and learn quite early. The descriptive statistics can be divided into two major types, descriptions of the central tendency and descriptions of how much the measurements vary, the variance. Descriptions of central tendency are the arithmetic mean, the mode, and the median. Descriptions of variance are the variance, the standard deviation, and the range.

dependent variables. Typically, the question one attempts to answer using statistics is that there is a relationship between two variables. To demonstrate that there is a relationship the experimenter must show that when one variable changes the second variable changes and that the amount of change is more than would be likely from mere chance alone.

There are two ways to figure the probability of an event. The first is to do a mathematical calculation to determine how often the event can happen. The second is to observe how often the event happens by counting the number of times the event could happen and also counting the number of times the event actually does happen.

The use of a mathematical calculation is when a person can say that the chance of the event rolling a one on a six sided die is one in six. The probability is figured by figuring the number of ways the event can happen and divide that number by the total number of possible outcomes. Another example is in a well shuffled deck of cards, what is the probability of the event of drawing a three. The answer is four in fifty two since there are four cards numbered three and there are a total of fifty two cards in a deck. The chance of the event of drawing a card in the suite of diamonds is thirteen in fifty two (there are thirteen cards of each of the four suites). The chance the event of drawing the three of diamonds is one in fifty two.

Sometimes, the size of the total event space, the number of different possible events, is not known. In that case, you will need to observe the event system and count the number of times the event actually happens versus the number of times it could happen but doesn't.

For instance, a warranty for a coffee maker is a probability statement. The manufacturer calculates that the probability the coffee maker will stop working before the warranty period ends is low. The way such a warranty is calculated involves testing the coffee maker to calculate how long the typical coffee maker continues to function. Then the manufacturer uses this calculation to specify a warranty period for the device. The actual calculation of the coffee maker's life span is made by testing coffee makers and the parts that make up a coffee maker and then using probability to calculate the warranty period.

Experiments, Outcomes and Events
The easiest way to think of probability is in terms of experiments and their potential outcomes. Many examples can be drawn from everyday experience: On the drive home from work, you can encounter a flat tire, or have an uneventful drive; the outcome of an election can include either a win by candidate A, B, or C, or a runoff.

Definition: The entire collection of possible outcomes from an experiment is termed the sample space, indicated as $$\Omega$$ (Omega)

The simplest (albeit uninteresting) example would be an experiment with only one possible outcome, say $$A$$. From elementary set theory, we can express the sample space as follows:

$$\Omega = \{ A \} $$

A more interesting example is the result of rolling a six sided dice. The sample space for this experiment is:

$$\Omega = \{ 1,2,3,4,5,6 \}$$

We may be interested in events in an experiment.

Definition: An event is some subset of outcomes from the sample space

In the dice example, events of interest might include a) the outcome is an even number b) the outcome is less than three

These events can be expressed in terms of the possible outcomes from the experiment: a) : $$ \{2,4,6\} $$ b) : $$ \{ 1,2 \}$$

We can borrow definitions from set theory to express events in terms of outcomes. Here is a refresher of some terminology, and some new terms that will be important later: $$ \cup $$ represents the Union of two events $$ \cap $$ represents the Intersection of two events $$\{\cdots\}^{c}$$ represents the complement of an event. For instance, "the outcome is an even number" is the complement of "the outcome is an odd number" in the dice example. $$ A \backslash B $$ represents difference, that is, $$A$$ but not $$B$$. For example, we may be interested in the event of drawing the queen of spades from a deck of cards. This can be expressed as the event of drawing a queen, but not drawing a queen of hearts, diamonds or clubs. $$\varnothing$$ or $$\{\}$$ represent an impossible event $$\Omega$$ represents a certain event $$A$$ and $$B$$ are called disjoint events if $$A\cap B = \varnothing$$

Probability
Now that we know what events are, we should think a bit about a way to express the likelihood of an event occurring. The classical definition of probability comes from the following. If we can perform our experiment over and over in a way that is repeatable, we can count the number of times that the experiment gives rise to event $$A$$. We also keep track of the number of times that we perform the same experiment. If we repeat the experiment a large enough number of times, we can express the probability of event $$A$$ as follows: $$P(A) = \frac{N_{A}}{N} $$ where $$N_{A}$$ is the number of times event $$A$$ occurred, and $$N$$ is the number of times the experiment was repeated. Therefore, the equation can be read as "the probability of event $$A$$ equals the number of times event $$A$$ occurs divided by the number of times the experiment was repeated (or the number of times event $$A$$ could have occurred)." As $$N$$ approaches infinity, the fraction above approaches the true probability of the event $$A$$. The value of $$P(A)$$ is clearly between 0 and 1. If our event is the certain event $$\Omega$$, then for each time we perform the experiment, the event $$\Omega$$ is observed; $$N_{\Omega} = N$$ and $$P(\Omega)=1$$. If our event is the impossible event $$\varnothing$$, we know $$N_{\varnothing}=0$$ and $$ P(\varnothing) = 0$$.

If $$A$$ and $$B$$ are disjoint events, then whenever event $$A$$ is observed, then it is impossible for event $$B$$ to be observed simultaneously. For example, a number may either be odd or even, so the sets of odd and even numbers from 1 to 20 are disjoint events. Therefore, the number of times events $$A$$ union $$B$$ occurs are equal to the number of times event $$A$$ occurred plus the number of times $$B$$ occurs. This can be expressed as: $$N(A\cup B) = N(A) + N(B)$$ Given our definition of probability, we can arrive at the following: $$P(A\cup B) = P(A) + P(B)$$

At this point it's worth remembering that not all events are disjoint events. For events that are not disjoint, we end up with the following probability definition. $$ P(A\cup B) = P(A) + P(B) - P(A\cap B)$$ How can we see this from example? Well, let's consider drawing from a deck of cards. I'll define two events: "drawing a Queen", and "drawing a Spade". It is immediately clear that these are not disjoint events, because you can draw a queen that is also a spade. There are four queens in the deck, so if we perform the experiment of drawing a card, putting it back in the deck and shuffling (what statisticians refer to as sampling with replacement, we will end up with a probability of $$\frac{1}{13}$$ for a queen draw. By the same argument, we obtain a probability for drawing a spade as $$\frac{1}{4}$$.  The expression $$P(A\cup B)$$ here can be translated as "the chance of drawing a queen or a spade".  If we incorrectly assume that for this case $$P(A\cup B) = P(A) + P(B)$$, we can simply add our probabilities together for "the chance of drawing a queen or a spade" as $$\frac{1}{13}+\frac{1}{4}$$.  If we were to gather some data experimentally, we would find that our results would differ from the prediction—the probability observed would be slightly less than $$\frac{1}{13}+\frac{1}{4}$$.  Why?  Because we're counting the queen of spades twice in our expression, once as a spade, and again as a queen. We need to count it only once, as it can only be drawn with probability of $$\frac{1}{52}$$. If we counted it twice with $$\frac{1}{13}+\frac{1}{4}$$, we will have to subtract it once: $$\frac{1}{13}+\frac{1}{4} - \frac{1}{52} $$. This reduces to $$\frac{4}{13}$$. Still confused?

Proof: If $$A$$ and $$B$$ are not disjoint, we have to avoid the double counting problem by exactly specifying their union. $$A \cup B = A \cup (B \backslash A) $$ so $$P(A \cup B) = P(A \cup (B \backslash A)) $$ $$A$$ and $$B \backslash A$$ are disjoint sets. We can then use the definition of disjoint events from above to express our desired result: $$P(A \cup B) = P(A) + P(B \backslash A)$$ We also know that $$P(B \backslash A) = P(B) - P(B\cap A)$$ so $$ P(A \cup B) = P(A) + P(B) - P(B\cap A)$$ Whew! Our first proof. I hope that wasn't too dry.

Conditional Probability
Many events are conditional on the occurrence of other events. Sometimes this coupling is weak. One event may become more or less probable depending on our knowledge that another event has occurred. For instance, the probability that your friends and relatives will call asking for money is likely to be higher if you win the lottery. In my case, I don't think this probability would change.
 * See also Probability/Conditional Probability

To obtain a formal definition of conditional probability, we begin with a working definition of probability: $$P(A) = \frac{N_{A}}{N}$$ Consider an additional event $$B$$, and a situation where we are only interested in the probability of the occurrence of $$A$$ when $$B$$ occurs. A way at this probability is to perform a set of experiments (trials) and only record our results (that is: occurrence of $$A$$) when the event $$B$$ occurs. In other words $$ \frac{N_{A\cap B}}{N_{B}} $$ We can divide through on top and bottom by $$N$$ the total number of trials to get $$P(A\cap B)/P(B)$$. We define this as 'conditional probability': $$P(A|B) = \frac{P(A\cap B)}{P(B)}$$ which when spoken, takes the sound "probability of $$A$$ given $$B$$."

Bayes' Law
An important theorem in statistics is Bayes' Law, which states that $$ P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$, It is easy to prove. We start with identical expressions for $$P(A\cap B)$$. We know that: $$ P(A\cap B) = P(B\cap A) $$, $$ P(A\cap B) = \frac{P(A|B)}{P(A)}$$, and $$ P(B\cap A) = \frac{P(B|A)}{P(B)}$$. Since $$ P(A\cap B) = P(B\cap A) $$, $$\frac{P(A|B)}{P(A)} = \frac{P(B|A)}{P(B)}$$. A Simple rearrangement of above line gives us Bayes' Law.

Independence
Two events $$A$$ and $$B$$ are called independent if the occurrence of one has absolutely no effect on the probability of the occurrence of the other. Mathematically, this is expressed as: $$P(A\cap B) = P(A)P(B)$$.

Random Variables
It's usually possible to represent the outcome of experiments in terms of integers or real numbers. For instance, in the case of conducting a poll, it becomes a little cumbersome to present the outcomes of each individual respondent. Let's say we poll ten people for their voting preferences (Republican - R, or Democrat - D) in two different electorial districts. Our results might look like this: $$\{RRRDRRDRRR\}$$ and $$\{DDDDDRDDDD\}$$ But we're probably only interested in the overall breakdown in voting preference for each district. If we assign an integer value to each outcome, say 0 for Democrat and 1 for Republican, we can obtain a concise summary of voting preference by district simply by adding the results together.

Discrete and Continuous Random Variables
There are two important subclasses of random variables: discrete random variable (DRV) and continuous random variable (CRV). Discrete random variables take only countably many values. It means that we can list the set of all possible values that a discrete random variable can take, or in other words, the number of possible values in the set that the variable can take is finite. If the possible values that a DRV X can take are a0,a1,a2,...an, the probability that X takes each is p0=P(X=a0), p1=P(X=a1), p2=P(X=a2),...pn=P(X=an). All these probabilities are greater than or equal zero.

For continuous random variables, we cannot list all possible values that a continuous variable can take because the number of values it can take is extremely large. It means that there is no use to calculate the probability of each value separately because the probability that the variable takes a particular value is extremely small and can be considered zero P(X=x)=0).

Expected Value
Expected value is a generalization of the concept of "average" to included "weighted averages". It is often denoted as E(X), but the overline and bracket notation is also used.

Example : Consider the following calculation of the average value of X if Xi take on these three values {1, 3, 3, 5}. Using the overline to denote "average" (or "mean" or "expectation value"), we have:

 * $$E(X) = \langle X \rangle = \overline X = \frac {\sum_{i=1}^N X_i}{N} =\frac{1+3+3+5}{4}$$

We can also group these by the number of times each number appears, using $$w_i$$ to denote the "weight", or number of times each value of Xi is repeated.

 * $$\overline X = \frac {\sum_{i=1}^N w_i\cdot X_i}{N} =\frac{1\cdot 1+2\cdot 3+ 1\cdot 5}{N} =\frac{12}{4}=3$$

Factor the N inside the sum:

 * $$\overline X = \sum_{i=1}^N \frac{w_i}{N}\cdot X_i =\frac 1 4 \cdot 1+\frac 2 4\cdot 3+ \frac 1 4\cdot 5$$

Suppose you randomly selected one of the values in the set, {1, 3, 3, 5}. The probability of selecting a 3 is 50% because 2 out of four of the elements equal 3. Letting P denote probability, we have P(3) = 2/4 = 50%, and P(1) = P(5) = 1/4. Defining the probability,

 * $$\overline X = \sum_{i=1}^N P_i \cdot X_i$$  where   $$P_i = \frac{w_i}{N}$$

With video lectures

 * 1) Probability and Random Processes, Mrityunjoy Chakraborty, IIT Kharagpur, 2004.
 * 2) Probability Primer, mathematicalmonk, YouTube
 * 3) Probabilistic Systems Analysis and Applied Probability, John Tsitsiklis, Massachusetts Institute of Technology, Fall 2010

Without videos lectures

 * 1) Fundamentals of Probability, David Gamarnik and John Tsitsiklis, Massachusetts Institute of Technology, Fall 2008.
 * 2) Applied Probability and Stochastic Modeling, Edward Ionides, University of Michigan, Fall 2013
 * 3) Probability and Statistics I, Motoya Machida, Tennessee Tech University, Fall 2010
 * 4) Probability and Statistics II, Motoya Machida, Tennessee Tech University, Spring 2011
 * 5) Probability and Random Processes, Tara Javidi, University of California, San Diego, 2012.
 * 6) Introduction to Statistical Methods, Joseph C. Watkins, University of Arizona, Spring 2013
 * 7) Probability Theory, Jenny Baglivo, Boston College, Fall 2013.
 * 8) Mathematical Statistics, Jenny Baglivo, Boston College, Fall 2013.
 * 9) Modern Statistics, Jenny Baglivo, Boston College, Spring 2014
 * 10) Probability, Colin Rundel, Duke University, Spring 2012.
 * 11) Applied Probability, Matthias Winkel, Oxford University, 2012.
 * 12) Probability, Frank King, University of Cambridge, 2007–08.
 * 13) Nonparametric Methods, Tom Fletcher, Utah University, Spring 2009.
 * 14) Communication Theory, B. Srikrishna, IIT Madras, Jul-Nov 2007.

Formula sheets

 * 1) Probability Formula Sheet by Colin Rundel
 * 2) Theoretical Computer Science Cheat Sheet
 * 3) Probabilistic Systems Analysis Cheat Sheet by Gowtham Kumar
 * 4) Probability and Statistics Cookbook by Matthias Vallentin, 2012
 * 5) Review of Measure Theory by Joel Feldman, 2008
 * 6) A Probability Refresher by Vittorio Castelli.
 * 7) Cheat Sheet for Probability Theory by Ingmar Land, 2005
 * 8) Combinatorics Cheat Sheet by Lenny Pitt

Other resources

 * 1) Probability and Statistics, Wolfram MathWorld
 * 2) e-Statistics, Dept of Maths, Tennessee Tech University.
 * 3) Online Statistics Education by Rice University (Lead Developer), University of Houston Clear Lake, and Tufts University
 * 4) Category:Probability theory paradoxes, Wikipedia
 * 5) Category:Probability Theory, ProofWiki.
 * 6) Notes on Financial Engineering, K. Sigman, Columbia University.
 * 7) Probability by Surprise by Susan Holmes
 * 8) GameTheory.net
 * 9) Java Applets provided by James W. Hardin
 * 10) Java applets for power and sample size by Russ Lenth
 * 11) Web Interface for Statistics Education at Claremont Graduate University


text:
Talk:Mathematics software

what about derive?

Algosim
Another free mathematical application is Algosim: https://algosim.org. --158.174.98.95 (discuss) 17:47, 23 July 2021 (UTC)

title: Talk:Mathematics software


text:
Talk:Probability and statistics

Introductory comments about "Wikiversity"
Okay, I'm sure this page will qualify me as a bonafied freak, especially now that I've gone around and looked at the other "Wikiversity" pages and seen how skeletal they were. But you know what? I really hope the day comes that we turn Education into a free resources that can be given to the masses rather than savored by the privileged few. I now trying to learn Spanish and hopefully in another two years or so I can start translating materials into that language.

So I was pretty interested in "riffing" on the idea of what a real Wikiversity would entail. Right now the Wikibooks are not looking much different from the Wikipedia articles which are great for what they are but there's a huge difference between amassing a large body of information where people can find it and teaching which involves:

 * patiently leading the student down a certain thought process
 * creating assignments that drill, repeat, test
 * finding ways of "explaining" things to a person who doesn't initially understand

These are the sorts of things a textbook (or "schoolbook") should focus on, and that'll take some planning. And for a topic as involved in Statistics, you'll have to impart a good foundation in the early materials and keep pushing forward until the advanced stuff. So I created a roadmap for what I believe would take a person of minimal age 18 (maybe as early as 16 for an intrepid prodigy) and take him or her through four years of hard work to the other side.

I know the chances of this actually taking solid root are pretty minimal, but you never know, do you?

--Murraytodd 06:09, 16 Sep 2004 (UTC)

This looks promising
Just a note of encouragement, I'm looking forward to more updates and perhaps contributing something myself.

Why is this here
There appears to be another book about statistics.

http://en.wikibooks.org/wiki/Statistics

Why are there two places for statistics?

Any of you know anything about this related section
Take a look at http://en.wikiversity.org/wiki/Topic:Statistics

Is this section active? --Lucas Gallindo 02:10, 11 September 2006 (UTC)

PS.: Nice job. What is your background on the subject? --

Steve Harris 22:06, 26 September 2009 (UTC) This is my first time on "Wikiversity" - I hope someone can help me - I want to know how bookmakers such as William Hill, Ladbrokes and Paddy Power etc., make a "round book" in horseracing. For example today's race - the 16.15 at Ascot - 4 runners, their prices were: 8/13, 5/2, 15/2 and 18/1. I believe that by adding these 4 fractions together the total comes to 107.5% - in other words the bookmaker would make 7.50 profit for every £100 he took in bets - what I want to know is how does he adjust his "book" each time someone puts on a bet? If someone could help me I would be very grateful. Steve Harris 22:06, 26 September 2009 (UTC)

title: Talk:Probability and statistics


text:
School talk:Mathematics/Philosophy of Mathematics

Top Structure
Suggestion for Top-level structure:

Basis: Proof

Supplying Mathematics

 * I Understanding Maths (Meta Maths)
 * Questioning Job (Axioms)
 * Questioning Scientific Program (Sentence)
 * Questioning World (Theory)
 * Questioning Universe (Formal System)
 * Questioning Maths?

 * II Doing Maths ( Processor .... 'Homo Mathematicus' .... Scientific Community )
 * Sign Processing / QA
 * Clarity (Who can understand it)
 * Scientific Community (Altruism)
 * 'Homo Scientificus'-based Model (Interaction amongst them)
 * Multi Community Model (Maths + others)

Applying Mathematics

 * III Understanding Application (Application to the real world (and beyond ?))
 * What is the real world like?
 * What Theory to choose on the Maths side?
 * How to apply the Theory chosen?
 * How 'good' can this approach be ( compared to others )

 * IV Doing Application ( The Process )
 * Elements
 * Players ( Processor ... 'Homo Mathematicus'... Opportunist )
 * Interactions ( stand alone ... virtual ... restricted )
 * Areas ( Science, Engineering, ... )
 * Models
 * Stand alone Models
 * Virtual (unrestricted) Communities
 * Restricted Communities
 * Restrictions in the real World
 * Models of Communities

= Collection of detailed topics =

Conversely - from a bottom-up view - the following terms and names should be contained in the above (distributed all over the course):

 * Aesthetics
 * Set theory
 * Logic
 * Georg Cantor & Infinity
 * Kurt Gödel & Completeness & Negation-Incopleteness
 * Gödel's Incompleteness Theorem
 * Continuum Hypothesis
 * Cardinality
 * Eugene Wigner & Effectiveness
 * Peano Axioms and Peano Arithmetics
 * Hermann Weyl & Cognition
 * Scepticism
 * Truth & Tarski
 * Concept & Frege

On Structuring
Questions: The 'bright' blackbox view: Mathematician also 'generates' understanding during his work.
 * How to write it down?
 * What formal concepts of Proof do exist?
 * How can I be sure that my Proof is correct?
 * Efficiency: how can I check it practically
 * Effectiveness: what can i be sure of/ how sure can I be - at all?
 * There are several ways to proof it - which one is the best?
 * Two Proofs are said to be equal iff ... (theoretical & practical answer)
 * How can I structure my Proof? (Lemmata, Corollaries & co.)
 * Are there Proof-Quality-Metrics?

Questions: Enhanced input view: Mathematician uses already existing work as input.
 * Is the Proof my only result? What to do with the understanding of the Theory I have developed during my work? How can I transfer this?
 * Why should I transfer it? Is this still science?
 * What to transfer?
 * For whom to transfer?
 * How to transfer?
 * Does this have an effect on the above questions?
 * Shell I seperate e.g. the question for the best way to prove sth from the transfer questions?
 * Does understanding help me with assuring the correctness?
 * Is there a notion of a formal/understandable Proof? (In practice it is, but is this just laziness or better than pure formal approaches?)

Questions: Introducing Definitions (belongs to bright blackbox view?)
 * What existing Theorems may I use?
 * The proved ones, but who decides this?
 * Have I to check all their proofs myself?
 * If not, who can I rely on - the scientific 'community'?

Questions:
 * What Definitions to introduce?
 * What is a Definition?
 * How is it defined?
 * How can I write Definitions down easily?
 * What is it good for?
 * For formality?
 * For understanding?
 * How to decide on 'define or not define'?
 * What to define always?
 * What to define never?
 * Are there good/bad examples?

More Questions: ...
 * What are the properties of the Theory? (sound, complete, ...) e.g. can I find a Proof at all?
 * Is the underlying Logic (concept of Proof) the right one?

Questions concerning the scientific program: ...
 * What Theorems to approach? (esp. does it make sense to prove this Theorem at all?)
 * What is the role of this Theory amongst all other Theories?


text:
Philosophy of mathematics/Georg Cantor

According to modern set theory, originally based on Cantor's seminal ideas, the union of two infinite disjoint classes, such as the odd and even natural numbers, has the same number of members as either constituent class. This violates the logical principle that the union of two disjoint classes is a distinct class, with a distinct cardinal number. The resulting theory of infinite classes is, therefore, constructed at the cost of abandoning this and related logical principles.

This casting aside of readily understood 'intuitive' principles in favour of a system of axioms and definitions constitutes the very foundation of modern mathematics, and largely precludes any challenge to the basis of Cantorian and subsequent theories of infinite classes or numbers. The danger of such a state of affairs is that the foundations of mathematics cannot be radically altered, thus precluding any advances which might result from a reappraisal of Cantor's theory of infinity.

An intuitive illustration
While this might seem "counter-intuitive" at first, a simple thought experiment can provide us with a feel for Cantor's theorem.

The "trick" is to simply count all of the odd (or even) natural numbers:

 * 1 → 1
 * 2 → 3
 * 3 → 5
 * 4 → 7
 * n → 2n-1
 * n+1 → 2(n+1) -1
 * n+1 → 2(n+1) -1

What you'll realise over time (when you're tired of counting) is that you'll never run out of odd numbers to count, and that for any odd number n, it will appear at (n+1)/2 in the counting order.

A notable exception to this rule is that a set made up of infinitely many infinite sets will in fact be larger than any of the member sets, according to Hilbert's paradox of the Grand Hotel.

title: Philosophy of mathematics/Georg Cantor


text:
Philosophy of mathematics/Gödel's Incompleteness Theorem

Austrian mathematician Kurt Gödel showed that any mathematical system complicated enough to include arithmetic must be incomplete. What does this mean? Simply that there are theorems in the system that are true but cannot be proven true from within the system. It may be possible to prove the theorem from outside the system. A lot of nonsense has been written about this topic by nonmathematicians, including assertions that we can never know anything. Some mathematical systems ARE complete, including ordinary geometry and propositional logic.

What Godel showed is that we can never have a complete knowledge of mathematics, and in the 1930's he shook the math world at its core when he proved this. Any Mathematical theory has axioms from which everything in the theory is derived. It is impossible to prove the axioms of a theory by using the theory, you have to go to a more fundamental theory to prove the axioms of the first theory. The problem is that you then have to go to an even more fundamental theroy to prove the axioms of the fundamental theory. The problem with that is you have to go to a more ...... ad infinitum. We can never prove the fundamental axioms and laws of mathematics on which everything is based and thus our knowledge will necessarily always be incomplete.


text:
Philosophy of Mathematics/Cardinality

Given the set {10, 56, 29, 72}, I ask you: how many elements are in this set?

In order to answer this question, you likely looked at the element "10" and thought "1", for "56", "2", and so on, until you ended up with the set {1, 2, 3, 4}. From this you concluded that the set had four elements.

Recall a common construction for the natural numbers: and so on.
 * ∅ is 0
 * {∅} is 1
 * {∅, {∅}} is 2
 * {∅, {∅}, {∅, {∅}}} is 3

This construction is convienent precisely because the number of elements, that is, the 'cardinality' of the set is identical to our intuiton of what a natural number is.

So then, the symbols |A|=|B| mean that there exists a one-to-one mapping from A onto B.

{10, 56, 29, 72} |   |   |   |  v   v   v   v { 1,  2,  3,  4}

This is the meaning of the notion of cardinality.

Infinite Sets
It seems obvious that we can find a cardinality for any finite set. But what of infinite sets?

The procedure for proving two infinite sets have the same cardinality involves setting up a mapping between the two sets and proving that it is one-to-one.

For example: Prove |Q|=|N|, if Q is the set of rational numbers and N is the set of natural numbers.

Recall that the rational numbers are the set of all numbers in the form $$-1^n \frac{a}{b}$$

Since n, a, and b are all natural numbers, we can use a Gödel numbering scheme to map each rational number onto a natural number:

$$f(n, a, b): N^3 \rightarrow N = 2^n 3^a 5^b$$

And such a mapping is one-to-one by the fundamental theorem of arithmetic.

This is a rather powerful result. It should be clear that any set whose elements can be totally expressed as a finite set of natural numbers has the same cardinality as that of the natural numbers. Because of this, the cardinality of the natural numbers has a special name, $$\aleph_0$$ (Aleph-Null). Sets that have cardinality $$\aleph_0$$ are called countable or denumerable.

Some sets have even greater cardinality than N, however. The set of real numbers, for instance, is even greater.

title: Philosophy of Mathematics/Cardinality