1787 US Constitution: during good behavior
1776 US Declaration of Independence: grievences
1773 John Adams, Works vol 3, p 522: during good behavior & quamdiu se bene gesserint*
1701 Act of Settlement: quamdiu se bene gesserint, patent for judges
1641 House of Lords petition: quamdiu se bene gesserint
1600 General Assembly of the Kirk of Scotland: ad vitam or ad culpam**
1465 De Officiis first printed
44BC Cicero writes De Officiis. Book II, i, contains quam diu and a form of gerere (to conduct)
* so long as they shall conduct themselves well. Note that gesserint is future perfect indicative. The future perfect is used in Latin conditional clauses.
** for life or until fault (see end of 1st session)
Thursday, October 21, 2010
Some Selections from Cicero's De Officiis (On Duties)
Beneficence and liberality:
Let me now, as I propossed, speak of beneficence and liberality, virtues that are the most agreeable to the nature of man, but which involve many precautionary considerations. For, in the first place, we are to take care lest our kindness should hurt both those whom it is meant to assist, and others. In the next place, it ought not to exceed our abilities; and it ought to be rendered to each in proportion to his worth. This is the fundamental standard of justice to which all these things should be referred. For they who do kindnesses which prove of disservice to the person they pretend to oblige, should not be esteemed beneficent nor generous, but injurious sycophants. And they who injure one party in order to be liberal to another, are guilty of the same dishonesty as if they should appropriate to themselves what belongs to another. (Offices I, xiv, p. 25)
Administration of government:
All who hope to rise in a state ought strictly to observe two rules of Plato. The first is, that they so keep in view the advantage of their fellow-citizens as to have reference to it in whatever they do, regardless of their individual interest. The second is, that their cares be applied to the whole of the state, lest while they are cherishing one part they abandon the others. For the adminsitration of government, like a guardianship, ought to be directed to the good of those who confer, and not of those who receive the trust. Now, they who consult the interests of one part of a community and neglect another, introduce into the state the greatest of all evils, sedition and discord. (Offices, I, xxv, p.44)
Magistrates:
Neither is it foreign to my purpose to touch upon the duties of magistrates, of private citizens, and of strangers. It is then the peculiar duty of a magistrate to bear in mind that he represents the state, and that he ought, therefore, to maintain its dignity and glory, to preserve its constitution, to act by its laws, and to remember that these things are committed to his fidelity. (Offices, I, xxxiv, p. 61)
Knowledge and protecting rights:
...For the knowledge and contemplation of nature is in a manner lame and unfinished, if it is followed by no activity; now activity is most perspicuous when it is exerted in protecting the rights of mankind.
It therefore has reference to the social interests of the human race, and is for that reason preferable to knowledge; and this every virtuous man maintains and exhibits in practice. For who is so eager in pursuing and examining the nature of things, that if, while he is handling and contemplating the noblest objects of knowledge, the peril and crisis of his country is made known to him, and that it is in his power to assist and relieve her, whould not intantly abandon and fling from him all those studies, even though he would be enabled to number the stars, or measure the dimensions of the world? And he would to the same were the safety of a friend or a parent concerned or endangered. From this consideration I infer, that the duties of justice are preferable to the studies and duties of knowledge, relating as they do to the interests of the human race, to which no anterior consideration ought to exist in the mind of man. (Offices I, xliii, p. 72)
Discourse versus thought:
Nor do those philosophers only instruct and educate those who are desirous of learning while alive and present among us; but they continue to do the same after death, by the monuments of their learning; for they neglect no point that relates to the constitution, the manners and the morals of their country; so that it appears as if they had dedicated all thir leisure to our advantage. Thus while they are themselves devoted to the studies of learning and wisdom, they make their understanding and their skill chiefly available to the service of mankind. It is therefore more serviceable to the public for a man to discourse copiously, provided it is to the purpose, than for a man to think ever so accurately without the power of expression; the reason is, because thought terminates in itself alone, but discourse affects those with whom we are connected in a community.
...Every duty therefore that operates of the good of human community and society, is preferable to that duty which is limited to speculation and knowledge. (Offices I, xliv, pp. 75-6)
Let me now, as I propossed, speak of beneficence and liberality, virtues that are the most agreeable to the nature of man, but which involve many precautionary considerations. For, in the first place, we are to take care lest our kindness should hurt both those whom it is meant to assist, and others. In the next place, it ought not to exceed our abilities; and it ought to be rendered to each in proportion to his worth. This is the fundamental standard of justice to which all these things should be referred. For they who do kindnesses which prove of disservice to the person they pretend to oblige, should not be esteemed beneficent nor generous, but injurious sycophants. And they who injure one party in order to be liberal to another, are guilty of the same dishonesty as if they should appropriate to themselves what belongs to another. (Offices I, xiv, p. 25)
Administration of government:
All who hope to rise in a state ought strictly to observe two rules of Plato. The first is, that they so keep in view the advantage of their fellow-citizens as to have reference to it in whatever they do, regardless of their individual interest. The second is, that their cares be applied to the whole of the state, lest while they are cherishing one part they abandon the others. For the adminsitration of government, like a guardianship, ought to be directed to the good of those who confer, and not of those who receive the trust. Now, they who consult the interests of one part of a community and neglect another, introduce into the state the greatest of all evils, sedition and discord. (Offices, I, xxv, p.44)
Magistrates:
Neither is it foreign to my purpose to touch upon the duties of magistrates, of private citizens, and of strangers. It is then the peculiar duty of a magistrate to bear in mind that he represents the state, and that he ought, therefore, to maintain its dignity and glory, to preserve its constitution, to act by its laws, and to remember that these things are committed to his fidelity. (Offices, I, xxxiv, p. 61)
Knowledge and protecting rights:
...For the knowledge and contemplation of nature is in a manner lame and unfinished, if it is followed by no activity; now activity is most perspicuous when it is exerted in protecting the rights of mankind.
It therefore has reference to the social interests of the human race, and is for that reason preferable to knowledge; and this every virtuous man maintains and exhibits in practice. For who is so eager in pursuing and examining the nature of things, that if, while he is handling and contemplating the noblest objects of knowledge, the peril and crisis of his country is made known to him, and that it is in his power to assist and relieve her, whould not intantly abandon and fling from him all those studies, even though he would be enabled to number the stars, or measure the dimensions of the world? And he would to the same were the safety of a friend or a parent concerned or endangered. From this consideration I infer, that the duties of justice are preferable to the studies and duties of knowledge, relating as they do to the interests of the human race, to which no anterior consideration ought to exist in the mind of man. (Offices I, xliii, p. 72)
Discourse versus thought:
Nor do those philosophers only instruct and educate those who are desirous of learning while alive and present among us; but they continue to do the same after death, by the monuments of their learning; for they neglect no point that relates to the constitution, the manners and the morals of their country; so that it appears as if they had dedicated all thir leisure to our advantage. Thus while they are themselves devoted to the studies of learning and wisdom, they make their understanding and their skill chiefly available to the service of mankind. It is therefore more serviceable to the public for a man to discourse copiously, provided it is to the purpose, than for a man to think ever so accurately without the power of expression; the reason is, because thought terminates in itself alone, but discourse affects those with whom we are connected in a community.
...Every duty therefore that operates of the good of human community and society, is preferable to that duty which is limited to speculation and knowledge. (Offices I, xliv, pp. 75-6)
Sunday, October 17, 2010
Plato and Cicero
Plato's Sophist is interesting because it illustrates the dialectical analyisis and the use of opposites. It's one of the better dialogues that I have read and I am a little disappointed that it was not in the book of Plato's dialogues that I have. As to the Stranger being a god, that is probably hyperbole although I suspect that Zeno played Son of Hermes when he and Parmenides visited Athens and met with Socrates. The Sophist probably draws from the philosophy of the Pythagoreans rather than that of that of the Athenians.
Cicero's De Officiis which discusses ones moral duty in serving society is good reading. Cicero had an inclination for philosophy and studied in Athens. The book is addressed to his son. Marcus Aurelius and others have done similar works. De Officiis was the second book to appear in print. It later served as a primary introduction to Latin. James Madison, who studied at Princeton, probably read him as part of his course of study. Thomas Jefferson had several works by Cicero in his library which were later donated to the Congressional Library. Much of what Cicero says is still relavent today.
Both Plato and Cicero wrote works discussing the republic. Much of Cicero's De Re Publica was concealed as a palimpsest until 1822 so it was not available at the time of the US Constitutional Convention.
Cicero's De Officiis which discusses ones moral duty in serving society is good reading. Cicero had an inclination for philosophy and studied in Athens. The book is addressed to his son. Marcus Aurelius and others have done similar works. De Officiis was the second book to appear in print. It later served as a primary introduction to Latin. James Madison, who studied at Princeton, probably read him as part of his course of study. Thomas Jefferson had several works by Cicero in his library which were later donated to the Congressional Library. Much of what Cicero says is still relavent today.
Both Plato and Cicero wrote works discussing the republic. Much of Cicero's De Re Publica was concealed as a palimpsest until 1822 so it was not available at the time of the US Constitutional Convention.
Wednesday, September 22, 2010
Impressions vs Forms
We have seen that we can find fits that are very close to the actual characteristics for diodes. But fits bear the same relation to characteristics as an image does to an object. Our observations are not perfect and there are distortions present. The same relation exists between a print and the typeset page from which it was made. It is impression versus form.
The ancient Greek philosophers assigned a greater reality to the forms than our perceptions of them. The problem is similar to that of doing fits with the patterns being somewhat imperfect. But the same pattern may be applicable to a number of similar objects (in our case diodes) with different coefficients in the characteristic equations.
In The Sophist Plato uses an "Eleatic Stranger" with extraordinary skill to try to capture what we mean by a sophist. The Stranger first uses the dialect method in an attempt to pin down the sophist but he proves to be an elusive quarry. He appears to be many things at once. After studying "being" and "not-being" and what we mean by these terms he comes to the conclusion that the sophist is a pretender. He does not possess or impart true knowledge. The dialectic is a process of defining a pair of cases to which the object in question either belongs or does not in order to specify what it is. The Stranger looks down on the Sophist and is cutting him at the same time. The Sophist does not rank with the Philosopher.
The case is not quite so bad with the fits but one still can't trust them completely. Theory might help narrow down equations to be fitted but measurements usually have errors associated with them and can produce unavoidable residuals as a result.
The ancient Greek philosophers assigned a greater reality to the forms than our perceptions of them. The problem is similar to that of doing fits with the patterns being somewhat imperfect. But the same pattern may be applicable to a number of similar objects (in our case diodes) with different coefficients in the characteristic equations.
In The Sophist Plato uses an "Eleatic Stranger" with extraordinary skill to try to capture what we mean by a sophist. The Stranger first uses the dialect method in an attempt to pin down the sophist but he proves to be an elusive quarry. He appears to be many things at once. After studying "being" and "not-being" and what we mean by these terms he comes to the conclusion that the sophist is a pretender. He does not possess or impart true knowledge. The dialectic is a process of defining a pair of cases to which the object in question either belongs or does not in order to specify what it is. The Stranger looks down on the Sophist and is cutting him at the same time. The Sophist does not rank with the Philosopher.
The case is not quite so bad with the fits but one still can't trust them completely. Theory might help narrow down equations to be fitted but measurements usually have errors associated with them and can produce unavoidable residuals as a result.
Saturday, September 18, 2010
5.1 V Zener Diode Conductance and Estimated Error in Voltages
We can refer to the ratio of small changes in current for small changes in voltage as a conductance*. It is the slope of the a line tangent to the V-I characteristics and in calculus is referred to as a derivative. The unit for conductance is the Siemens, S. If one plots this function for the 5.1 V Zener diode one sees that the conductance starts to increase rapidly at about 4.75 volts.
This conductance can be used to estimate the errors in the voltage measurements and compute corrections. The formula is ΔV = ΔI/G. The results for the steep part of the V-I characterists is shown below. This shows that even though the residuals are relatively large the corrections to the voltage needed to place the points on the curve are within about 1/2 the precision of the measurements and appear to be just round off errors.
The residuals for these points were more randomly distributed. The "corrections" for lower voltages were larger where the curve was flatter and where the fit appeared to deviate somewhat from the data. The conclusion is that relatively larger residuals are to be expected where the slope of the curve is relatively steep.
*edit: This is not the usual conductance which is defined in terms of Ohm's Law, G = I/V. My terminology is somewhat lax. A more appropriate expression might be "small signal conductance" or "differential conductance."
This conductance can be used to estimate the errors in the voltage measurements and compute corrections. The formula is ΔV = ΔI/G. The results for the steep part of the V-I characterists is shown below. This shows that even though the residuals are relatively large the corrections to the voltage needed to place the points on the curve are within about 1/2 the precision of the measurements and appear to be just round off errors.
The residuals for these points were more randomly distributed. The "corrections" for lower voltages were larger where the curve was flatter and where the fit appeared to deviate somewhat from the data. The conclusion is that relatively larger residuals are to be expected where the slope of the curve is relatively steep.
*edit: This is not the usual conductance which is defined in terms of Ohm's Law, G = I/V. My terminology is somewhat lax. A more appropriate expression might be "small signal conductance" or "differential conductance."
Thursday, September 16, 2010
Does Least Squares Make the Best Choice for Coefficients?
Least Squares is often used to find a fit to a set of data. Linear Least Squares is easier since the solution for the unknown coefficients only involves a set of equations which are linear functions of the unknowns. The primary condition for a least squares fit is that the sum of the squares of the residuals be a minimum. The result is that portions of the fitted curve can have a definite deviation from the data set and still result in a minimum value for the sum.
For the fits of the data for the Zener diode I used Mathcad's genfit function. For the composite functions of the last fit the individual fits can also be done by fitting lines in a log-log plot*. One of the problems that I encountered in doing the fits is that the fitted function tended to cut corners in the region of the "knee" where the characteristics change from lower currents to higher ones. As a result there is a correlation among neighboring residuals there. Intuitively one would expect that the residuals would be randomly distributed with respect to the fitted curve for the best fit. There seemed to be something wrong with the residuals.
I decided to give the rms residual more control in doing the fit. I tried dividing the data into sections with each section being represented by a term in the fit function. The result is that the residuals differed for each region. This method also seemed somewhat arbitrary since the terms for the function depended on how the data was partitioned. The results suggested the use of a Taylor series for the current involving exponential functions. Fitting a quadratic function of the first fit didn't work. The difference between the fit for lower currents and the data looked closer to an exponential function that was fairly linear for lower currents so I tried multiplying the current by a factor involving an exponential term. That worked much better than the Taylor series. Varying the number of data points included in the first fit optimized the final fit. The rms residual was closer to the result of a tedious process involving a search for the coefficients in a region of fixed size and a random choice from among a few of the lowest values of the sum of the squares of the residuals to avoid getting "stuck in a loop." (The results of the second fit method weren't consistent. The coefficients posted were the best values found.)
The results show the utility of the residuals in analyzing the results of a fit and comparing one fit method with another.
The economists like to use linear models for optimization problems but nonlinear models may prove to be more useful. Fits can be used to find approximate functions for observable relations in a theory. Doing what is best for the individual is not necessisarily at odds with doing what is best for the group. The problem may be similar to that of finding the best fit for a set of data.
*edit: One would have to do a semi-log plot. For the first fit involving the lower curve one would also have to ignore the small constant, u_0, and plot log(I1/mA) vs V. For the second plot it would be log(I2/I1 - 1) vs I1 (I2>I1) with I1 being the current in the first fit and I2 that of the second fit.
For the fits of the data for the Zener diode I used Mathcad's genfit function. For the composite functions of the last fit the individual fits can also be done by fitting lines in a log-log plot*. One of the problems that I encountered in doing the fits is that the fitted function tended to cut corners in the region of the "knee" where the characteristics change from lower currents to higher ones. As a result there is a correlation among neighboring residuals there. Intuitively one would expect that the residuals would be randomly distributed with respect to the fitted curve for the best fit. There seemed to be something wrong with the residuals.
I decided to give the rms residual more control in doing the fit. I tried dividing the data into sections with each section being represented by a term in the fit function. The result is that the residuals differed for each region. This method also seemed somewhat arbitrary since the terms for the function depended on how the data was partitioned. The results suggested the use of a Taylor series for the current involving exponential functions. Fitting a quadratic function of the first fit didn't work. The difference between the fit for lower currents and the data looked closer to an exponential function that was fairly linear for lower currents so I tried multiplying the current by a factor involving an exponential term. That worked much better than the Taylor series. Varying the number of data points included in the first fit optimized the final fit. The rms residual was closer to the result of a tedious process involving a search for the coefficients in a region of fixed size and a random choice from among a few of the lowest values of the sum of the squares of the residuals to avoid getting "stuck in a loop." (The results of the second fit method weren't consistent. The coefficients posted were the best values found.)
The results show the utility of the residuals in analyzing the results of a fit and comparing one fit method with another.
The economists like to use linear models for optimization problems but nonlinear models may prove to be more useful. Fits can be used to find approximate functions for observable relations in a theory. Doing what is best for the individual is not necessisarily at odds with doing what is best for the group. The problem may be similar to that of finding the best fit for a set of data.
*edit: One would have to do a semi-log plot. For the first fit involving the lower curve one would also have to ignore the small constant, u_0, and plot log(I1/mA) vs V. For the second plot it would be log(I2/I1 - 1) vs I1 (I2>I1) with I1 being the current in the first fit and I2 that of the second fit.
Impossible Choices
Arrow's Social Choice and Individual Values and Sen's Liberal Paradox point out some of the failings of the democratic process and the impossibility of reaching a group concensus on a number of options. But this just validates the need for a mechanism to make a decision in the absence of a majority or the situation where the outcome is unclear. In the case of a tie in the US Senate the Vice President gets the deciding vote. Congress is empowered to decide the outcome of the elections of its members. In these situations the decision that is made is not democratic since the voters failed to reach a decision.
If the choice is left solely to individuals though some people can end up benefiting at the expense of others. The most powerful factions end up being self-serving and disregard the interests of others.
Arrow also points out that capital is not the only way to improve productivity. Technological innovation can substitute for capital. This includes the arrangement of the workplace. Improved economy has its benefits. There are other ways of improving. One is education which improves the value of workers. And an economic slowdown may indicate that there are forces at work which tend to obstruct progress.
If the choice is left solely to individuals though some people can end up benefiting at the expense of others. The most powerful factions end up being self-serving and disregard the interests of others.
Arrow also points out that capital is not the only way to improve productivity. Technological innovation can substitute for capital. This includes the arrangement of the workplace. Improved economy has its benefits. There are other ways of improving. One is education which improves the value of workers. And an economic slowdown may indicate that there are forces at work which tend to obstruct progress.
Subscribe to:
Posts (Atom)