Space Math: Solar Storms and You! Exploring Satellite Design

This educator’s guide includes activities in space science. Educational levels: Intermediate elementary, Middle school

Space Math IV

This booklet includes 31 problems that involve a variety of math skills, including scientific notation, simple algebra, and calculus. Topics covered include black holes, ice on mercury, a mathematical model of the solar interior, sunspots, the heliopause, and coronal mass ejections, among many others. (8.5 x11, 28 pages, 11 color images, PDF file) Educational levels: Intermediate elementary, Middle school, High school

Thinking Impossible Things

“There is no use in trying,” said Alice; “one can’t believe impossible things.” “I dare say you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half an hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast”. Lewis Carroll, Through the Looking Glass. It is a rather common view among philosophers that one cannot, properly speaking, be said to believe, conceive, imagine, hope for, or seek what is impossible. Some philosophers, for instance George Berkeley and the early Wittgenstein, thought that logically contradictory propositions lack cognitive meaning (informational content) and cannot, therefore, be thought or believed. Philosophers who do not go as far as Berkeley and Wittgenstein in denying that impossible propositions or states of affairs are thinkable, may still claim that it is impossible to rationally believe an impossible proposition. On a classical “Cartesian” view of belief, belief is a purely mental state of the agent holding true a proposition p that she “grasps” and is directly acquainted with. But if the agent is directly acquainted with an impossible proposition, then, presumably, she must know that it is impossible. But surely no rational agent can hold true a proposition that she knows is impossible. Hence, no rational agent can believe an impossible proposition. Thus it seems that on the Cartesian view of propositional attitudes as inner mental states in which proposition are immediately apprehended by the mind, it is impossible for a rational agent to believe, imagine or conceive an impossible proposition. Ruth Barcan Marcus (1983) has suggested that a belief attribution is defeated once it is discovered that the proposition, or state of affairs that is believed is impossible. According to her intuition, just as knowledge implies truth, belief implies possibility. It is commonplace that people claim to believe propositions that later turn out to be impossible. According to Barcan Marcus, the correct thing to say in such a situation is not: I once believed that A but I don’t believe it any longer since I have come to realize that it is impossible that A. What one should say is instead: It once appeared to me that I believed that A, but I did not, since it is impossible that A. Thus, Barcan Marcus defends what we might call Alice’s thesis: Necessarily, for any proposition p and any subject x, if x believes p, then p is possible. Alice’s thesis that it is impossible to hold impossible beliefs, seems to come into conflict with our ordinary practices of attributing beliefs. Consider a mathematical example. Some mathematicians believe that CH (the continuum hypothesis) is true and others believe that it is false. But if CH is true, then it is necessarily true; and if it is false, then it is necessarily false. Regardless of whether CH is true or false, the conclusion seems to be that there are mathematicians who believe impossible propositions. Examples of apparent beliefs in impossible propositions outside of mathematics are also easy to come by. Consider, for example, Kripke’s (1999) story of the Frenchman Pierre who without realizing it has two different names ‘London’ and ‘Londres’ for the same city, London. After having arrived in London, Pierre may assent to ‘Londres is beautiful and London is not beautiful’ without being in any way irrational. It seems reasonably to infer from this that Pierre believes that Londres is beautiful and London is not beautiful. But since ‘Londres’ and ‘London’ are rigid designators for the same city, it seems to follow from this that Pierre believes the inconsistent proposition that we may express as ‘London is both beautiful and not beautiful’

Space Math: Lunar Math

This booklet includes 17 problems relating to the Moon and its exploration. Images from NASA are analyzed to determine image scales and the physical sizes of various crates and features. The probability of meteor impacts near a lunar colony are calculated, and the horizon distance is determined using simple geometry. Also covered are: determining the mass of the Moon, a simple model for the lunar interior, heat flow rates, extracting oxygen from lunar rock, and lunar transits and eclipses. (8.5 x11, 28 pages, 11 color images, PDF file) Educational levels: Intermediate elementary, Middle school, High school

Paradoxes of Demonstrability

In this paper I consider two paradoxes that arise in connection with the concept of demonstrability, or absolute provability. I assume—for the sake of the argument—that there is an intuitive notion of demonstrability, which should not be conflated with the concept of formal deducibility in a (formal) system or the relativized concept of provability from certain axioms. Demonstrability is an epistemic concept: the rough idea is that a sentence is demonstrable if it is provable from knowable basic (“self-evident”) premises by means of simple logical steps. A statement that is demonstrable is also knowable and a statement that is actually demonstrated is known to be true. By casting doubt upon apparently central principles governing the concept of demonstrability, the paradoxes of demonstrability presented here tend to undermine the concept itself—or at least our understanding of it. As long as we cannot find a diagnosis and a cure for the paradoxes, it seems that the coherence of the concepts of demonstrability and demonstrable knowledge are put in question. There are of course ways of putting the paradoxes in quarantine, for example by imposing a hierarchy of languages a` la Tarski, or a ramified hierarchy of propositions and propositional functions a` la Russell. These measures, however, helpful as they may be in avoiding contradictions, do not seem to solve the underlying conceptual problems. Although structurally similar to the semantic paradoxes, the paradoxes discussed in this paper involve epistemic notions: “demonstrability”, “knowability”, “knowledge”... These notions are “factive” (e.g., if A is demonstrable, then A is true), but similar paradoxes arise in connection with “nonfactive” notions like “believes”, “says”, “asserts”.3 There is no consensus in the literature concerning the analysis of the notions involved—often referred to as “propositional attitudes”—or concerning the treatment of the paradoxes they give rise to

Space Math: Black Hole Math

This booklet includes 11 problems of increasing difficulty that explore the properties of black holes. Students learn about the relationship between black hole size and mass, time dilation, energy extraction, accretion disks, and what’s inside a black hole by working with a series of math problems that feature simple algebra, scientific notation, exponential functions, and the Pythagorean Theorem. (8.5 x11, 28 pages, 11 color images, PDF file) Educational levels: Intermediate elementary, Middle school, High school

Space Math: Hinode Math

This booklet is a collection of 15 problems that incorporate data from the Hinode solar observatory. These problems cover topics in sunspot structure, spectroscopy, solar rotation, magnetic fields, density and temperature of hot gases, and solar flares. The math skills covered include finding the scale of an image to determine actual physical sizes in images, time calculations, volumes of cylinders, graph analysis, and scientific notation. Educational levels: Intermediate elementary, Middle school, High school

P is not equal to NP

SAT is not in P, is true and provable in a simply consistent extension B' of a first order theory B of computing, with a single finite axiom characterizing a universal Turing machine. Therefore, P is not equal to NP, is true and provable in a simply consistent extension B" of B.Comment: In the 2nd printing the proof, in the 1st printing, of theorem 1 is divided into three parts a new lemma 4, a new corollary 8, and the remaining part of the original proof. The 2nd printing contains some simplifications, more explanations, but no error has been correcte

The Auroral Battery and Electrical Circuit

In this activity students use simple algebra and basic electrical principles to explore the formation of auroras by currents from the magnetotail region. They discover that scientists have proposed that changes in the magnetic field in the magnetotail region cause releases of energy that eventually supply the "battery" (stored energy) to light up the aurora on Earth. Students also learn that energy is stored in a magnetic field, and the amount depends on how strong the field is, and how big a volume it occupies. During the activity they estimate just how much magnetic energy is available in the magnetic tail region of Earth. Educational levels: High school