Mathematical beliefs

One of the most interesting aspects of Schoenfeld s research is his focus on mathematical beliefs and their role in problem solving. He argues that students have beliefs that lie between the purely affective and the purely cognitive domains of human behavior. Included in a student s belief system are misconceptions, preconceptions, and intuitions about events in the world. These may be at odds with new, incoming instruction and may create... 

Finally, the one-belief position makes belief too easy. It is relatively easy to acquire the belief that the sea contains a natural kind that also is found in lakes, falls from the sky, is potable, and so on. It is relatively easy to tag this kind water. (In so tagging it we are not, of course, assuming that the kind always has the properties it has when we come across it in lakes etc.) But what we have just described is roughly how things were before Lavoisier, and on the one-belief position, we thereby believed that the sea contains H2O. It was that easy. What is more, this belief was a rationally supported one (because the belief that the sea contains water was and they are one and the same belief). All that can be said to be hard on the one-belief position was the discovery of what belief a certain sentence in our mouths, namely, The sea contains water, expresses. It took Lavoisier s experiments to tell us which belief we expressed in certain words. It is common to criticize Stalnaker s metalinguistic account of mathematical beliefs as underselling the manifest interest of mathematical discovery. This would seem a more extreme underplaying of Lavoisier s achievement. 

Speaking to young Flory that day, Carothers said in his slow and measured way, You know, the polymer field is an area where it is my belief [that] mathematics could be applied. Flory spent the rest of his life doing just that and dominated the field of polymers for two decades after World War II. So gentle was Carothers leadership, however, that it was only after his death that Flory realized how much of a shield he had been, and how much of an influence.. .. It was an extraordinary opportunity, I realize now in retrospect. ... 

The core of the Evidence theory lies in the combination of belief assignments. As already described, several belief combination rules exist, each of them corresponding to an interpretation of the conflict between bba. As a consequence, a rule should be first chosen according to the interpretation of conflict appropriate to the final objective. In addition, some mathematical considerations should be also accounted for. Indeed, only the Dempster s and Smets rules are associative. This means that for other combination rules e.g., Yager s or Dubois and Trade s rules), we have (i) either to perform a simultaneous combination of all available bba or (ii) to determine the sequence of combinations appropriate to the final objective. The first choice is satisfactory since it does not need to perform any assumption on the combination order. However, it implies to compute a great number of intersections of focal elements and is thus difficult to apply for more than seven bba (due to the computation time required). 

The second caveat is that there is a price to be paid for extending the paradox of self-deception down the scale of the paradox of irrationality. The extension starts with the discovery that a self-deceiver who persuades himself that p does not have to believe not-p. When this is accepted, as it must be, a gap begins to open up between self-deception and the deception of others. If A had, and was able to evaluate premises that made p logically or mathematically impossible, and yet he told that p, we would probably say that he had deceived , but the case is less clear when A only has, and is able to evaluate inductive evidence against p. Even in this case he has deceived in an indirect way about his evidence, but it is not clear that he has deceived him about what he actually told him, unless he himself has formed the belief that not-p. If this were the only divergence between self-deception and the deception of others, we might make an adjustment to the concept of self-deception in order to get rid of it. In fact, as we shall soon see, there are other divergences. 

There are two kinds of case that present this problem. If the subject accepts the wishful mismonitoring of his feelings or valuations or intentions, it seems that he cannot do so consciously, because that would require him to be aware that not-p, while accepting the belief that p. The other possibility is that a logically valid argument or a mathematically correct computation may leave him no latitude to form... 

The idea of hidden variables is fairly common in chemical models such as the kinetic gas model. This theory is formulated in terms of molecular momenta that remain hidden, and evaluated against measurements of macroscopic properties such as pressure, temperature and volume. Electronic motion is the hidden variable in the analysis of electrical conduction. The firm belief that hidden variables were mathematically forbidden in quantum systems was used for a long time to discredit Bohm s ideas. Without joining the debate it can be stated that this proof has finally been falsified. 

Bacon s belief that the principles of mathematics and number underlined everything in the world led to his discovery that the calendar was in need of reform his suggestions were finally accepted in 1582 when the Gregorian system, as it was known, was finally adopted. It also led to his creating one of the earliest maps of the world. (Now rumoured to be in the Vatican library.)... 

Belief in impossibility is the starting point for logic, deductive mathematics, and natural science. It can originate in a mind that has freed itself from belief in its own omnipotence. 

The belief that the underlying order of the world can be expressed in mathematical form lies at the very heart of science. So deep does this belief run that a branch of science is considered not to be properly understood until it can be cast in mathematics. 

The worst, i.e., most dangerous, feature of accepting the null hypothesis is the giving up of explicit uncertainty... Mathematics can sometimes be put in such black-and-white terms, but our knowledge or belief about the external world never can. 

We have already indicated our belief that the modular wave-hierarchy we have elaborated may be the system that can supply mathematical rigor to the idea of the Coordinative Conditions. 

As a result of inadequate resources and the belief that applied engineering held the key, and seeking to establish a rational and consistent basis for making decisions, EPA personnel soon adopted a judgment-based technique that often relied heavily on mathematical models for calculating risks quantitative risk assessment. Somewhat later the assessment and management of risks were conceptually separated, though the two activities have always remained interdependent. 

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