Hello, this is Hans van der Kras senior lecturer at IT Delft Institute for Water Education. In this video I'm going to show how to use lookup tables with PC Raster in QGIS. Let's say we want to select only the houses class out of the buildg Raster here, then we can simply use the raster calculator, we don't need lookup tables, and this also to demonstrate that you can use the PC Raster maps with the raster calculator without any problem. So here I say when the buildg Raster equals value 1, which is for houses, then give me boolean true and otherwise give me boolean false. Now in the raster calculator it doesn't work if you use PC Raster as an output format, so stick to the geotiff. Then when I run it I get here a boolean layer, a geotiff with the houses. When I style it this becomes clear, so two classes. Let's add the label, zero is no houses and one is houses, and make the colors more intuitive because it's boolean. Let's make false white, so it looks like a background, and make true green. Now let's have a look at our road map. It has different types of roads, dirt road and tarmac, and I would like to reclassify it in boolean with road and no road, and that is very easy using a lookup table that you can simply make as an ASCII file in, for example, notepad. PC Raster uses a table in ASCII format where the first column is the original value from the raster, and the second is what it will be assigned, so in this case zero remains zero, one will be boolean true, and two will also be boolean true. I save the table as reclassroads. So I use a lookup operation, input is roads, then I load the lookup table, reclassroads, and I need to specify the output data type, in our case that's boolean, and I can save it, and let's call it isroad. Run it, and it creates a boolean layer which has cells which are no road and cells that are road. We can style it, we palette the unique values, we classify no road and the road, and use boolean red for false and green for true. But can we also classify ranges, because that would be very easy. Well, that's possible. Let's classify the build g layer into classes that are industry related and that are not industry related. So if I look at the legend, then I can see that classes zero, one, and two are no industry, and three, four, and five are industry related. So I can make a lookup table with ranges. Also that's a simple ASCII table. You can use the less than simple to say that it does not include that value of the range, and if we keep it open, it means all the values and we separate with a comma. So this means all the values less than two, including two, become boolean zero. And the second line means all the values from three, including three, until the highest value that it finds becomes boolean one, true. So I save it as reclass industry, and then I can use lookup, give the table the output again will be boolean, and I call it is industry, and I can run it and copy the style to make it a boolean look. There it is industry versus no industry. Just need to change the labels, but I'm not going to do that here. Set the projection for the question marks. Now we have only looked at boolean outputs, but what if we look at other types of outputs from reclassifications and those ranges. So let's reclassify our DTM layer. And what I'm going to do is specify three different ranges, everything smaller than 200 and 10 meters, not including 210 will become class one. Then from 210 and including 210 to 250, not including 250 becomes class two, and 250, including the 250, until the largest value that it finds becomes class number three. So that's an ordinal scale. Let's call it reclass DTM, run lookup, use the DTM as input and our new lookup table, reclass DTM. And the output is ordinal in this case. These are classes, but the order does matter, so then it becomes ordinal. Let's call it DTM ordinal. I run it, move it to the top, and now it needs proper styling. Let's also set the projection. Ordinal also classes, but then we need to use a ramp because the hierarchy of the order should be visible. So something like this. There's also a lookup linear tool, which interpolates values between the values that it finds in the table. If you click on the link, it will go to the page in the documentation of PC Raster. Here you see how it works. And here, for example, we have a table with input value 0257, and that will be an output 0110. And here we see an input of 1, which does not exist in the table, but then it takes the middle value, so it will be 0.5 in the result. So that's basically how it works with an interpolation. These were the lookup operations. I'm still working on making it possible to have multiple rasters as an input and use tables with multiple columns because that's also supported by PC Raster. But that's not ready yet. When it's ready, I'll post a video.


To convert ideas into reality, we might need a team with varied skills working and collaborating around different types of assets. My name is William and I want to show you an example of collaboration using GitLab. Consider this scenario. We have these folks where one of them is a designer. His day-to-day activities involve working and creating different types of assets, like wireframes, mockups, and high quality images. And in the same team, a developer who works all day long, mostly with text files or code. And we need these two folks to collaborate and create a landing page. Let's start this journey as a designer. I am the designer now. I want to kick off the landing page creation, so in GitLab I create a new issue. I give it a name, explain the goal, and assign it to the developer who will code the site. And once I submit the issue, then I click on Designs tab, and here I can upload high quality images, being one of them the website mockup. These assets are now under version control. Remember, in this story, right now I am the designer, and these images are the input for the developer to start working creating the site. Now in the team, we can start discussions on the designs by clicking on the image on the exact location you would like the discussion to be focused on. Alright, so now let's switch roles. And I am the developer now. In GitLab, I can see I have an issue assigned. I can also see the designs and understand better what's expected from me to code. But wait, I can see that there is one mockup and two images. It's not clear to me which one of those should I use. So in the discussion, I can ask this question to the designer and get his answer right there in the mockup of interest. Good, the designer clarified my doubts. Now as a developer, I can move on and start doing what I know best. From the same issue page in GitLab, I can create a branch and a merge request in one single click from where I can fire up the web IDE and without leaving GitLab work in the page code. Here I have uploaded the image assets to the repo. There you go. That's Han Solo. I can go on and change the image path, observe the diff in the code, and commit to the branch. Good, I have GitLab CI configured. So once I commit and the changes are merged, this will start my pipeline where I can see from here it succeeded. It was successful, so that's great. It means I didn't break anything while doing these changes. Let's see the result. Looks nice. It has the image the designer wanted to use. Okay, so wrapping up, in this quick example, the team could manage and collaborate around assets and files of different types. This collaboration fits in the development practices and patterns such as branching, version control, merge request, and so on. It made it easier to move faster, having communication available at every level. All of these capabilities can lead to more efficient teams in turning ideas into reality. Stay tuned and let's continue learning at GitLab.


File 35 of a Treatise of Human Nature by David Hume, Volume 2. This LibriVox recording is in the Pamlic domain, recording by George Yeager. Book 3 of Morals. Part 1 of Virtue and Vice in General. Section 1. Moral Distinctions Not Derived from Reason. There is an inconvenience which attends all abstruse reasoning that it may silence without convincing an antagonist and requires the same intense study to make us sensible of its force that was at first requisite for its invention. When we leave our closet and engage in the common affairs of life, its conclusions seem to vanish like the phantoms of the night on the appearance of the morning, and it is difficult for us to retain even that conviction which we had attained with difficulty. This is still more conspicuous in a long chain of reasoning where we must preserve to the end the evidence of the first propositions, and where we often lose sight of all the most received maxims, either of philosophy or common life. I am not, however, without hopes that the present system of philosophy will acquire new force as it advances, and that our reasonings concerning morals will corroborate whatever has been said concerning the understanding and the passions. Society is a subject that interests us above all others. We fancy the peace of society to be at stake in every decision concerning it, and it is evident that this concern must make our speculations appear more real and solid than where the subject is in a great measure indifferent to us. That affects us, we conclude, can never be a chimera, and as our passion is engaged on the one side or the other, we naturally think that the question lies within human comprehension, which in other cases of this nature we are apt to entertain some doubt of. Without this advantage I never should have ventured upon a third volume of such abstruse philosophy in an age wherein the greatest part of men seem agreed to convert reading into an amusement, and to reject everything that requires any considerable degree of attention to be comprehended. It has been observed that nothing is ever present to the mind but its perceptions, and that all the actions of seeing, hearing, judging, loving, hating, and thinking fall under this denomination. The mind can never exert itself in any action which we may not comprehend under the term of perception, and consequently that term is no less applicable to those judgments by which we distinguish moral good and evil than to every other operation of the mind. To approve of one character, to condemn another, are only so many different perceptions. Now as perceptions resolve themselves into two kinds, that is, impressions and ideas, this distinction gives rise to a question with which we shall open up our present inquiry concerning morals. Whether it is by means of our ideas or impressions, we distinguish betwixt vice and virtue, and pronounce an action blamable or praiseworthy. This will immediately cut off all loose discourses and declamations, and reduce us to something precise and exact on the present subject. Those who affirm that virtue is nothing but a conformity to reason, that there are eternal fitnesses and unfitnesses of things which are the same to every rational being that considers them, that the immutable measures of right and wrong impose an obligation not only on human creatures but also on the deity himself, all these systems concur in the opinion that morality, like truth, is discerned merely by ideas and by their juxtaposition and comparison. In order, therefore, to judge of these systems, we need only consider whether it be possible from reason alone to distinguish betwixt moral good and evil, or whether there must concur some other principles to enable us to make that distinction. If morality had naturally no influence on human passions and actions, it were in vain to take such pains to inculcate it, and nothing would be more fruitless than that multitude of rules and precepts with which all moralists abound. Philosophy is commonly divided into speculative and practical, and as morality is always comprehended under the latter division, it is supposed to influence our passions and actions, and to go beyond the calm and indolent judgments of the understanding. And this is confirmed by common experience, which informs us that men are often governed by their duties, and are deterred from some actions by the opinion of injustice, and impelled to others by that of obligation. Since morals, therefore, have an influence on the actions and affections, it follows that they cannot be derived from reason, and that, because reason alone, as we have already proved, can never have any such influence. Morals excite passions and produce or prevent actions. Reason of itself is utterly impotent in this particular. The rules of morality, therefore, are not conclusions of our reason. No one, I believe, will deny the justness of this inference, nor is there any other means of evading it than by denying that principle on which it is founded. As long as it is allowed that reason has no influence on our passions and actions, it is in vain to pretend that morality is discovered only by a deduction of reason. An active principle can never be founded on an inactive, and if reason be inactive in itself, it must remain so in all its shapes and appearances, whether it exerts itself in natural or moral subjects, whether it considers the powers of external bodies or the actions of rational beings. It would be tedious to repeat all the arguments by which I have proved, in Book 2, Part 3, Section 3, that reason is perfectly inert and can never either prevent or produce any action or affection. It will be easy to recollect what has been said upon that subject. I shall only recall on this occasion one of these arguments which I shall endeavor to render still more conclusive and more applicable to the present subject. Reason is the discovery of truth or falsehood. Truth or falsehood consists in an agreement or disagreement, either to the real relations of ideas or to real existence and matter of fact. Whatever, therefore, is not susceptible of disagreement or disagreement is incapable of being true or false and can never be an object of our reason. Now it is evident our passions, volitions, and actions are not susceptible of any such agreement or disagreement, being original facts and realities complete in themselves, and implying no reference to other passions, volitions, and actions. It is impossible, therefore, that they can be pronounced either true or false and be either contrary or conformable to reason. This argument is of double advantage to our present purpose, for it proves directly that actions do not derive their merit from a conformity to reason nor their blame from a contrarity to it, and it proves the same truth more indirectly by shooing us that as reason can never immediately prevent or produce any action by contradicting or approving of it, it cannot be the source of moral good and evil which are found to have that influence. Actions may be laudable or blameable, but they cannot be reasonable or unreasonable. Laudable or blameable, therefore, are not the same with reasonable or unreasonable. The merit and demerit of actions frequently contradict and sometimes control our natural propensities, but reason has no such influence. Moral distinctions, therefore, are not the offspring of reason. Reason is wholly inactive and can never be the source of so active a principle as conscience or a sense of morals. But perhaps it may be said that though no will or action can be immediately contradictory to reason, yet we may find such a contradiction in some of the attendance of the action, that is, in its causes or effects. The action may cause a judgment or may be obliquely caused by one when the judgment concurs with a passion, and by an abusive way of speaking which philosophy will scarce allow of, the same contrarity may, upon that account, be ascribed to the action. How far this truth or falsehood may be the source of morals it will now be proper to consider. It has been observed that reason in a strict and philosophical sense can have an influence on our conduct only after two ways, either when it excites a passion by informing us of the existence of something which is a proper object of it, or when it discovers the connection of causes and effects so as to afford us means of exerting any passion. These are the only kinds of judgment which can accompany our actions or can be said to produce them in any manner, and it must be allowed that these judgments may often be false and erroneous. A person may be affected with passion by supposing a pain or pleasure to lie in an object which has no tendency to produce either of these sensations, or which produces the contrary to what is imagined. A person may also take false measures for the attaining his end and may retard by his foolish conduct instead of forwarding the execution of any project. These false judgments may be thought to affect the passions and actions which are connected with them, and may be said to render them unreasonable in a figurative and improper way of speaking. But though this be acknowledged, it is easy to observe that these errors are so far from being the source of all immorality that they are commonly very innocent and draw no matter of guilt upon the person who is so unfortunate as to fall into them. They extend not beyond a mistake of fact which moralists have not generally supposed criminal as being perfectly involuntary. I am more to be lamented than blamed if I am mistaken with regard to the influence of objects in producing pain or pleasure, or if I know not the proper means of satisfying my desires. No one can ever regard such errors as a defect in my moral character—a fruit, for instance, that is really disagreeable, appears to me at a distance. And through mistake I fancy it to be pleasant and delicious. Here is one error. I choose certain means of reaching this fruit which are not proper for my end. Here is a second error. Nor is there any third one which can ever possibly enter into our reasoning's concerning actions. I ask therefore if a man in this situation and guilty of these two errors is to be regarded as vicious and criminal, however unavoidable they might have been, or if it be possible to imagine that such errors are the sources of all immorality. And here it may be proper to observe that if moral distinctions be derived from the truth or falsehood of those judgments, they must take place wherever we form the judgments. Nor will there be any difference whether the question be concerning an apple or a kingdom, or whether the error be avoidable or unavoidable. For as the very essence of morality is supposed to consist in an agreement or disagreement to reason, the other circumstances are entirely arbitrary and can never either bestow on any action the character of virtuous or vicious or deprived of that character, to which we may add that this agreement or disagreement not admitting of degrees, all virtues and vices would of course be equal. Should it be pretended that though a mistake of fact be not criminal, yet a mistake of right often is, and that this may be the source of immorality, I would answer that it is impossible such a mistake can ever be the original source of immorality, since it supposes a real right and wrong, that is, a real distinction in morals independent of these judgments. A mistake therefore of right may become a species of immorality, but it is only a secondary one and is founded on some other antecedent to it. As to those judgments which are the effects of our actions and which when false give occasion to pronounce the actions contrary to truth and reason, we may observe that our actions never cause any judgment either true or false in ourselves, and that it is only on others they have such an influence. It is certain that an action on many occasions may give rise to false conclusions in others, and that a person who through a window sees any lewd behavior of mine with my neighbor's wife may be so simple as to imagine she is certainly my own. In this respect my action resemble somewhat a lie or falsehood, only with this difference which is material, that I perform not the action with any intention of giving rise to a false judgment in another, but merely to satisfy my lust and passion. It causes however a mistake and false judgment by accident, and the falsehood of its effects may be ascribed by some odd figurative way of speaking to the action itself. But still I can see no pretext of reason for asserting that the tendency to cause such an error is the first spring or original source of all immorality. One might think it were entirely superfluous to prove this if a late author, William Wollaston, the religion of nature delineated, London, 1722, who has had the good fortune to obtain some reputation, had not seriously affirmed that such a falsehood is the foundation of all guilt and moral deformity. Yet we may discover the fallacy of his hypothesis, we need only consider that a false conclusion is drawn from an action only by means of an obscurity of natural principles which makes a cause be secretly interrupted in its operation by contrary causes, and renders the connection betwixt to objects uncertain and variable. Now as a like uncertainty and variety of causes take place even in natural objects, and produce a like error in our judgment, if that tendency to produce error were the very essence of vice and immorality, it should follow that even inanimate objects might be vicious and immoral. It is in vain to urge that inanimate objects act without liberty and choice. Whereas liberty and choice are not necessary to make an action produced in us an erroneous conclusion, they can be in no respect essential to morality, and I do not readily perceive upon this system how they can ever come to be regarded by it. If the tendency to cause error be the origin of immorality, that tendency and immorality would in every case be inseparable. Add to this that if I had used the precaution of shutting the windows while I indulged myself in those liberties with my neighbor's wife, I should have been guilty of no immorality, and that, because my action, being perfectly concealed, would have had no tendency to produce any false conclusion. For the same reason, a thief who steals in by a ladder at a window and takes all imaginable care to cause no disturbance is in no respect criminal, for either he will not be perceived, or if he be, it is impossible he can produce any error, nor will anyone from these circumstances take him to be other than what he really is. It is well known that those who are squint-sighted do very readily cause mistakes in others, and that we imagine they salute or are talking to one person while they address themselves to another. Are they, therefore, upon that account immoral? Besides, we may easily observe that in all those arguments there is an evident reasoning in a circle. A person who takes possession of another's goods and uses them as his own in a manner declares them to be his own, and this falsehood is the source of the immorality of injustice. But is property or right or obligation intelligible without an antecedent morality? A man that is ungrateful to his benefactor in a manner affirms that he never received any favors from him, but in what manner? Is it because it is his duty to be grateful? But this supposes that there is some antecedent rule of duty and morals. Is it because human nature is generally grateful and makes us conclude that a man who does any harm never received any favor from the person he harmed? But human nature is not so generally grateful as to justify such a conclusion. Or if it were, is an exception to a general rule in every case criminal for no other reason than because it is an exception. But what may suffice entirely to destroy this whimsical system is that it leaves us under the same difficulty to give a reason why truth is virtuous and falsehood vicious as to account for the merit or turpitude of any other action. I shall allow, if you please, that all immorality is derived from this supposed falsehood in action, provided you can give me any plausible reason why such a falsehood is immoral. If you consider rightly of the matter, you will find yourself in the same difficulty as at the beginning. This last argument is very conclusive, because if there be not an evident merit or turpitude annexed to this species of truth or falsehood, it can never have any influence upon our actions. For whoever thought of forbearing any action, because others might possibly draw false conclusions from it, or whoever performed any that he might give rise to true conclusions. And a footnote 12. Thus, upon the whole, it is impossible that the distinction betwixt moral good and evil can be made by reason, since that distinction has an influence upon our actions of which reason alone is incapable. Reason and judgment may indeed be the immediate cause of an action, by prompting or by directing a passion, but it is not pretended that a judgment of this kind, either in its truth or falsehood, is attended with virtue or vice. And as to the judgments which are caused by our judgments, they can still less bestow those moral qualities on the actions which are their causes. But to be more particular and to shoo that those eternal, immutable, fitnesses and unfitnesses of things cannot be defended by sound philosophy, we may weigh the following considerations. If the thought and understanding were alone capable of fixing the boundaries of right and wrong, the character of virtuous and vicious either must lie in some relations of objects or must be a matter of fact which is discovered by our reasoning. This consequence is evident, as the operations of human understanding divide themselves into two kinds, the comparing of ideas and the inferring of matter of fact, where virtue discovered by the understanding it must be an object of one of these operations, nor is there any third operation of the understanding which can discover it. There has been an opinion very industriously propagated by certain philosophers that morality is susceptible of demonstration, and though no one has ever been able to advance a single step in those demonstrations, yet it is taken for granted that this science may be brought to an equal certainty with geometry or algebra. Upon this supposition, vice and virtue must consist in some relations, since it is allowed on all hands that no matter of fact is capable of being demonstrated. Let us therefore begin with examining this hypothesis and endeavor, if possible, to fix those moral qualities which have been so long the objects of our fruitless researches. Point out distinctly the relations which constitute morality or obligation that we may know wherein they consist, and after what manner we must judge of them. If you assert that vice and virtue consist in relations susceptible of certainty and demonstration, you must confine yourself to those four relations which alone admit of that degree of evidence, and in that case you run into absurdities from which you will never be able to extricate yourself. For as you make the very essence of morality to lie in the relations, and as there is no one of these relations but what is applicable not only to an irrational but also to an inanimate object, it follows that even such objects must be susceptible of merit or demerit. Resemblance, contrarity, degrees in quality, and proportions in quantity and number. All these relations belong as properly to matter as to our actions, passions, and volitions. It is unquestionable, therefore, that morality lies not in any of these relations nor the sense of it in their discovery. Footnote 13. As a proof how confused our way of thinking on this subject commonly is, we may observe that those who assert that morality is demonstrable do not say that morality lies in the relations and that the relations are distinguishable by reason. They only say that reason can discover such an action in such relations to be virtuous and such another vicious. It seems they thought it sufficient if they could bring the word relation into the proposition without troubling themselves whether it was to the purpose or not. But here I think is plain argument. Demonstrative reason discovers only relations, but that reason, according to this hypothesis, discovers also vice and virtue. These moral qualities, therefore, must be relations. When we blame any action in any situation, the whole complicated object of action and situation must form certain relations wherein the essence of vice consists. This hypothesis is not otherwise intelligible. For what does reason discover when it pronounces any action vicious? Does it discover a relation or a matter of fact? These questions are decisive and must not be eluded. End of footnote 13. Should it be asserted that the sense of morality consists in the discovery of some relation distinct from these and that our enumeration was not complete when we comprehended all demonstrable relations under four general heads? To this I know not what to reply, till someone be so good as to point out to me this new relation. It is impossible to refute a system which has never yet been explained. In such a manner of fighting in the dark a man loses his blows in the air and often places them where the enemy is not present. I must, therefore, on this occasion rest contented with requiring the two following conditions of anyone that would undertake to clear up this system. First, as moral good and evil belong only to the actions of the mind and are derived from our situation with regard to external objects, the relations from which these moral distinctions arise must lie only betwixt internal actions and external objects, and must not be applicable either to internal actions compared among themselves or to external objects when placed in opposition to other external objects. For as morality is supposed to attend certain relations, if these relations could belong to internal actions considered singly, it would follow that we might be guilty of crimes in ourselves and independent of our situation with respect to the universe, and in like manner, if these moral relations could be applied to external objects, it would follow that even inanimate beings would be susceptible of moral beauty and deformity. Now it seems difficult to imagine that any relation can be discovered betwixt our passions, volitions, and actions compared to external objects which relation might not belong either to these passions and volitions or to these external objects compared among themselves. But it will be still more difficult to fulfill the second condition requisite to justify this system. According to the principles of those who maintain an abstract rational difference betwixt moral good and evil, and a natural fitness and unfitness of things, it is not only supposed that these relations being eternal and immutable are the same when considered by every rational creature, but their effects are also supposed to be necessarily the same, and it is concluded they have no less or rather a greater influence in directing the will of the deity than in governing the rational and virtuous of our own species. These two particulars are evidently distinct. It is one thing to know virtue and another to conform the will to it. In order therefore to prove that the measures of right and wrong are eternal laws obligatory on every rational mind, it is not sufficient to shoe the relations upon which they are founded. We must also point out the connection betwixt the relation and the will, and must prove that this connection is so necessary that in every well-disposed mind it must take place and have its influence, though the difference betwixt these minds be in other respects immense and infinite. Now besides what I have already proved, that even in human nature no relation can ever alone produce any action, besides this I say it has been shunned in treating of the understanding that there is no connection of cause and effect such as this is supposed to be, which is discoverable otherwise than by experience, and of which we can pretend to have any security by the simple consideration of the objects. All beings in the universe, considered in themselves, appear entirely loose and independent of each other. It is only by experience we learn their influence and connection, and this influence we ought never to extend beyond experience. Thus it will be impossible to fulfill the first condition required to the system of eternal rational measures of right and wrong, because it is impossible to shoe those relations upon which such a distinction may be founded, and it is as impossible to fulfill the second condition, because we cannot prove a priori that these relations, if they really existed and were perceived, would be universally forcible and obligatory. But to make these general reflections more clear and convincing, we may illustrate them by some particular instances wherein this character of moral good or evil is the most universally acknowledged. Of all crimes that human creatures are capable of committing, the most horrid and unnatural is in gratitude, especially when it is committed against parents, and appears in the more flagrant instances of wounds and death. This is acknowledged by all mankind, philosophers as well as the people. The question only arises among philosophers whether the guilt or moral deformity of this action be discovered by demonstrative reasoning or be felt by an internal sense and by means of some sentiment which the reflecting on such an action naturally occasions. This question will soon be decided against the former opinion if we can shoe the same relations in other objects without the notion of any guilt or iniquity attending them. Reason or science is nothing but the comparing of ideas and the discovery of their relations, and if the same relations have different characters, it must evidently follow that those characters are not discovered merely by reason. To put the affair therefore to this trial, let us choose any inanimate object such as an oak or elm, and let us suppose that by the dropping of its seed it produces a sapling below it, which springing up by degrees, at last overtops and destroys the parent tree. I ask if in this instance there be wanting any relation which is discoverable in parasite or in gratitude. Is not the one tree the cause of the other's existence, and the latter the cause of the destruction of the former in the same manner as when a child murders his parent? It is not sufficient to reply that a choice or will is wanting. For in the case of parasite a will does not give rise to any different relations, but is only the cause from which the action is derived, and consequently produces the same relations that in the oak or elm arrives from some other principles.  Here then the same relations have different causes, but still the relations are the same, and as their discovery is not in both cases attended with a notion of immorality, it follows that that notion does not arise from such a discovery. But to choose an instance still more resembling, I would faint ask anyone why incest in the human species is criminal, and why the very same action and the same relations in animals have not the smallest moral turpitude and deformity. If it be answered that this action is innocent in animals because they have not reason sufficient to discover its turpitude, but that man, being endowed with that faculty which ought to restrain him to his duty, the same action instantly becomes criminal to him. Should this be said, I would reply that this is evidently arguing in a circle. For before reason can perceive this turpitude, the turpitude must exist, and consequently is independent of the decisions of our reason and is their object more properly than their effect. According to this system then, every animal that has sense and appetite and will, it is, every animal must be susceptible of all the same virtues and vices for which we ascribe praise and blame to human creatures. All the difference is that our superior reason may serve to discover the vice or virtue, and by that means may augment the blame or praise. But still this discovery supposes a separate being in these moral distinctions, and a being which depends only on the will and appetite, and which, both in thought and reality, may be distinguished from the reason. Animals are susceptible of the same relations with respect to each other as the human species, and therefore would also be susceptible of the same morality if the essence of morality consisted in these relations. Their want of a sufficient degree of reason may hinder them from perceiving the duties and obligations of morality, but can never hinder these duties from existing, since they must antecedently exist in order to their being perceived. Reason must find them and can never produce them. This argument deserves to be weighed, as being, in my opinion, entirely decisive. Where does this reasoning only prove that morality consists not in any relations that are the objects of science? But if examined, we'll prove with equal certainty that it consists not in any matter of fact which can be discovered by the understanding. This is the second part of our argument, and if it can be made evident, we may conclude that morality is not an object of reason. But can there be any difficulty in proving that vice and virtue are not matters of fact whose existence we can infer by reason? Take any action allowed to be vicious, willful murder, for instance. Examine it in all lights, and see if you can find that matter of fact or real existence which you call vice. In whichever way you take it, you find only certain passions, motives, volitions, and thoughts. There is no other matter of fact in the case. The vice entirely escapes you as long as you consider the object. You never can find it till you turn your reflection into your own breast and find a sentiment of disapprobation which arises in you towards this action. Matter is a matter of fact, but it is the object of feeling, not of reason. It lies in yourself, not in the object. So that when you pronounce any action or character to be vicious, you mean nothing but that from the constitution of your nature you have a feeling or sentiment of blame from the contemplation of it. Peace and virtue, therefore, may be compared to sounds, colors, heat, and cold, which, according to modern philosophy, are not qualities in objects, but perceptions in the mind. And this discovery in morals, like that other in physics, is to be regarded as a considerable advancement of the speculative sciences, though, like that too, it has little or no influence on practice. Nothing can be more real or concernous more than our own sentiments of pleasure and uneasiness. And if these be favorable to virtue and unfavorable to vice, no more can be requisite to the regulation of our conduct and behavior. I cannot forbear adding to these reasonings an observation which may, perhaps, be found of some importance. In every system of morality which I have hitherto met with, I have always remarked that the author proceeds for some time in the ordinary way of reasoning and establishes the being of a God, or makes observations concerning human affairs, when of a sudden I am surprised to find that instead of the usual copulations of propositions is and is not, I meet with no proposition that is not connected with an ought or an ought-not. This change is imperceptible, but is, however, of the last consequence. For as this ought or ought-not expresses some new relation or affirmation, it is necessary that it should be observed and explained, and at the same time that a reason should be given for what seems altogether inconceivable how this new relation can be a deduction from others which are entirely different from it. But as authors do not commonly use this precaution, I shall presume to recommend it to the readers, and am persuaded that this small attention would subvert all the vulgar systems of morality and let us see that the distinction of vice and virtue is not founded merely on the relations of objects, nor is perceived by reason. End of File thirty-five File thirty-six of A Treatise of Human Nature, by David Hume, Volume two. This LibriVox recording is in the public domain, recording by George Yeager. Book three of morals, part one of virtue and vice in general. Section two, moral distinctions derived from a moral sense. Thus the course of the argument leads us to conclude that since vice and virtue are not discoverable merely by reason or the comparison of ideas, it must be by means of some impression or sentiment they occasion that we are able to mark the difference betwixt them. Our decisions concerning moral rectitude and depravity are evidently perceptions, and as all perceptions are either impressions or ideas, the exclusion of the one is a convincing argument for the other. Morality therefore is more properly felt than judged of, though this feeling or sentiment is commonly so soft and gentle that we are apt to confound it with an idea according to our common custom of taking all things for the same which have any near resemblance to each other. The next question is of what nature are these impressions, and after what manner do they operate upon us? Here we cannot remain long in suspense, but must pronounce the impression arising from virtue to be agreeable, and that proceeding from vice to be uneasy. Every moment's experience must convince us of this. There is no spectacle so fair and beautiful as a noble and generous action, nor any which gives us more abhorrence than one that is cruel and treacherous. No enjoyment equals the satisfaction we receive from the company of those we love and esteem, as the greatest of all punishments is to be obliged to pass our lives with those we hate or condemn. A very play or romance may afford us instances of this pleasure which virtue conveys to us and pain which arises from vice. Now since the distinguishing impressions by which moral good or evil is known are nothing but particular pains or pleasures, it follows that in all inquiries concerning these moral distinctions it will be sufficient to shoe the principles which make us feel a satisfaction or uneasiness from the survey of any character in order to satisfy us why the character is laudable or blameable. When action or sentiment or character is virtuous or vicious, why? Because its view causes a pleasure or uneasiness of a particular kind, in giving a reason therefore for the pleasure or uneasiness we sufficiently explain the vice or virtue. To have the sense of virtue is nothing but to feel a satisfaction of a particular kind from the contemplation of a character. The very feeling constitutes our praise or admiration. We go no farther, nor do we inquire into the cause of the satisfaction. We do not infer a character to be virtuous because it pleases, but in feeling that it pleases after such a particular manner we in effect feel that it is virtuous. The case is the same as in our judgments concerning all kinds of beauty and tastes and sensations. Our approbation is implied in the immediate pleasure they convey to us. I have objected to the system which establishes eternal rational measures of right and wrong that it is impossible to shoe in the actions of reasonable creatures any relations which are not found in external objects, and therefore if morality always attended these relations it were possible for inanimate matter to become virtuous or vicious. Now it may in like manner be objected to the present system that if virtue and vice be determined by pleasure and pain these qualities must in every case arise from the sensations and consequently any object whether animate or inanimate, rational or irrational might become morally good or evil provided it can excite a satisfaction or uneasiness. But though this objection seems to be the very same it has by no means the same force in the one case as in the other. For first it is evident that under the term pleasure we comprehend sensations which are very different from each other and which have only such a distant resemblance as is requisite to make them be expressed by the same abstract term. A good composition of music and a bottle of good wine equally produce pleasure and what is more their goodness is determined merely by the pleasure but shall we say upon that account that the wine is harmonious or the music of a good flavor. In like manner an inanimate object and the character or sentiments of any person may both of them give satisfaction but as the satisfaction is different this keeps our sentiments concerning them from being confounded and makes us ascribe virtue to the one and not to the other. Nor is every sentiment of pleasure or pain which arises from characters and actions of that peculiar kind which makes us praise or condemn. The good qualities of an enemy are hurtful to us but may still command our esteem and respect. It is only when a character is considered in general without reference to our particular interest that it causes such a feeling or sentiment as denominates it morally good or evil. It is true those sentiments from interest and morals are apt to be confounded and naturally run into one another. It seldom happens that we do not think an enemy vicious and can distinguish betwixt his opposition to our interest and real villainy or baseness. But this hinders not but that the sentiments are in themselves distinct and a man of temper and judgment may preserve himself from these illusions. In like manner though it is certain a musical voice is nothing but one that naturally gives a particular kind of pleasure yet it is difficult for a man to be sensible that the voice of an enemy is agreeable or to allow it to be musical. But a person of a fine ear who has the command of himself can separate these feelings and give praise to what deserves it. Secondly, we may call to remembrance the preceding system of the passions in order to remark a still more considerable difference among our pains and pleasures. Pride and humility, love and hatred are excited when there is anything presented to us that both bears a relation to the object of the passion and produces a separate sensation related to the sensation of the passion. Now virtue and vice are attended with these circumstances. They must necessarily be placed either in ourselves or others and excite either pleasure or uneasiness and therefore must give rise to one of these four passions which clearly distinguishes them from the pleasure and pain arising from inanimate objects that often bear no relation to us. And this is perhaps the most considerable effect that virtue and vice have upon the human mind. It may now be asked in general concerning this pain or pleasure that distinguishes moral good and evil from what principles is it derived and whence does it arise in the human mind? To this I reply first that it is absurd to imagine that in every particular instance these sentiments are produced by an original quality and primary constitution. For as the number of our duties is in a manner infinite, it is impossible that our original instincts should extend to each of them and from our very first infancy impress on the human mind all that multitude of precepts which are contained in the completest system of ethics. Such a method of proceeding is not conformable to the usual maxims by which nature is conducted where a few principles produce all that variety we observe in the universe and everything is carried on in the easiest and most simple manner. It is necessary therefore to abridge these primary impulses and find some more general principles upon which all our notions of morals are founded. But in the second place should it be asked whether we ought to search for these principles in nature or whether we must look for them in some other origin, I would reply that our answer to this question depends upon the definition of the word nature than which there is none more ambiguous and equivocal. If nature be opposed to miracles not only the distinction betwixt vice and virtue is natural but also every event which has ever happened in our world accepting those miracles on which our religion is founded. In saying then that the sentiments of vice and virtue are natural in this sense we make no very extraordinary discovery. That nature may also be opposed to rare and unusual and in this sense of the word which is the common one there may often arise disputes concerning what is natural or unnatural and one may in general affirm that we are not possessed of any very precise standard by which these disputes can be decided. Point and rare depend upon the number of examples we have observed and as this number may gradually increase or diminish it will be impossible to fix any exact boundaries betwixt them. We may only affirm on this head that if ever there was anything which could be called natural in this sense the sentiments of morality certainly may since there never was any nation of the world nor any single person in any nation who was utterly deprived of them and who never in any instance should the least approbation or dislike of manners. These sentiments are so rooted in our constitution and temper that without entirely confounding the human mind by disease or madness it is impossible to extirpate and destroy them. But nature may also be opposed to artifice as well as to what is rare and unusual and in this sense it may be disputed whether the notions of virtue be natural or not. We readily forget that the designs and projects and views of men are principles as necessary in their operation as heat and cold, moist and dry. But taking them to be free and entirely our own it is usual for us to set them in opposition to the other principles of nature. Would it therefore be demanded whether the sense of virtue be natural or artificial, I am of opinion that it is impossible for me at present to give any precise answer to this question. Perhaps it will appear afterwards that our sense of some virtues is artificial and that of others natural. The discussion of this question will be more proper when we enter upon an exact detail of each particular vice and virtue. FOOTNOTE 14 In the following discourse natural is also opposed sometimes to civil, sometimes to moral. The opposition will always discover the sense in which it is taken. END OF FOOTNOTE 14 Meanwhile it may not be amiss to observe from these definitions of natural and unnatural that nothing can be more unphilosophical than those systems which assert that virtue is the same with what is natural and vice with what is unnatural. For in the first sense of the word nature as opposed to miracles both vice and virtue are equally natural, and in the second sense as opposed to what is unusual perhaps virtue will be found to be the most unnatural. At least it must be owned that heroic virtue being as unusual is as little natural as the most brutal barbarity. As to the third sense of the word it is certain that both vice and virtue are equally artificial and out of nature. For however it may be disputed whether the notion of a merit or demerit in certain actions be natural or artificial it is evident that the actions themselves are artificial and are performed with a certain design and intention, otherwise they could never be ranked under any of these denominations. It is impossible therefore that the character of natural and unnatural can ever in any sense mark the boundaries of vice and virtue. Thus we are still brought back to our first position, that virtue is distinguished by the pleasure and vice by the pain that any action, sentiment or character gives us by the mere view and contemplation. This decision is very commodious because it reduces us to this simple question. Why any action or sentiment upon the general view or survey gives a certain satisfaction or uneasiness, in order to shoo the origin of its moral rectitude or depravity without looking for any incomprehensible relations and qualities which never did exist in nature nor even in our imagination by any clear and distinct conception. I flatter myself and I have executed a great part of my present design by a state of the question which appears to me so free from ambiguity and obscurity. And of File 36. File 37 of a Treatise of Human Nature by David Hume, Volume 2. This LibriVox recording is in the public domain, recording by George Yeager, Book 3 of Morals, Part 2 of Justice and Injustice, Section 1, Justice, Whether a Natural or Artificial Virtue. I have already hinted that our sense of every kind of virtue is not natural, but that there are some virtues that produce pleasure and approbation by means of an artifice or contrivance which arises from the circumstances and necessity of mankind. Of this kind I assert justice to be, and shall endeavour to defend this opinion by a short and I hope convincing argument before I examine the nature of the artifice from which the sense of that virtue is derived. It is evident that when we praise any actions we regard only the motives that produce them and consider the actions as signs or indications of certain principles in the mind and temper. The external performance has no merit. We must look within to find the moral quality. Thus we cannot do directly and therefore fix our attention on actions as on external signs, but these actions are still considered as signs, and the ultimate object of our praise and approbation is the motive that produced them. After the same manner, when we require any action or blame a person for not performing it, we always suppose that one in that situation should be influenced by the proper motive of that action and we esteem it vicious in him to be regardless of it. If we find upon inquiry that the virtuous motive was still powerful over his breast, though checked in its operation by some circumstances unknown to us, we retract our blame and have the same esteem for him as if he had actually performed the action which we require of him. It appears therefore that all virtuous actions derive their merit only from virtuous motives and are considered merely as signs of those motives. From this principle I conclude that the first virtuous motive which bestows a merit on any action can never be a regard to the virtue of that action but must be some other natural motive or principle. To suppose that the mere regard to the virtue of the action may be the first motive which produced the action and rendered it virtuous is to reason in a circle. Before we can have such a regard, the action must be really virtuous and this virtue must be derived from some virtuous motive and consequently the virtuous motive must be different from the regard to the virtue of the action. A virtuous motive is requisite to render an action virtuous. An action must be virtuous before we can have a regard to its virtue. Some virtuous motive therefore must be antecedent to that regard. Nor is this merely a metaphysical subtlety but enters into all our reasonings in common life, though perhaps we may not be able to place it in such distinct philosophical terms. We blame a father for neglecting his child. Why? Because it shoes a want of natural affection, which is the duty of every parent. We're not natural affection a duty, the care of children could not be a duty, and it were impossible we could have the duty in our eye in the attention we give to our offspring. In this case therefore all men suppose a motive to the action distinct from a sense of duty. Here is a man that does many benevolent actions, relieves the distressed, comforts the afflicted, and extends his bounty even to the greatest strangers. No character can be more amiable and virtuous. We regard these actions as proofs of the greatest humanity. This humanity bestows a merit on the actions. A regard to this merit is, therefore, a secondary consideration. And derived from the antecedent principle of humanity, which is meritorious and laudable. In short, it may be established as an undoubted maxim that no action can be virtuous or morally good unless there be in human nature some motive to produce it distinct from the sense of its morality. But may not the sense of morality or duty produce an action without any other motive? I answer it may, but this is no objection to the present doctrine. When any virtuous motive or principle is common in human nature, a person who feels his heart devoid of that motive may hate himself upon that account, and may perform the action without the motive from a certain sense of duty in order to acquire by practice that virtuous principle, or at least to disguise to himself as much as possible his want of it. A man that really feels no gratitude in his temper is still pleased to perform grateful actions and thinks he has by that means fulfilled his duty. Actions are at first only considered as signs of motives, but it is usual in this case, as in all others, to fix our attention on the signs and neglect in some measure the thing signified. But though on some occasions a person may perform an action merely out of regard to its moral obligation, yet still this supposes in human nature some distinct principles which are capable of producing the action and whose moral duty renders the action meritorious. Now to apply all this to the present case I suppose a person to have lent me a sum of money on condition that it be restored in a few days, and also suppose that after the expiration of the term agreed on he demands the sum. I ask, what reason or motive have I to restore the money? It will perhaps be said that my regard to justice and abhorrence of villainy and navery are sufficient reasons for me if I have the least grain of honesty or sense of duty and obligation. And this answer no doubt is just and satisfactory to man in his civilized state, and when trained up according to a certain discipline and education. But in his rude and more natural condition, if you are pleased to call such a condition natural, this answer would be rejected as perfectly unintelligible and sophisticated. For one in that situation would immediately ask you, wherein consists this honesty and justice which you find in restoring alone and abstaining from the property of others. It does not surely lie in the external action. It must therefore be placed in the motive from which the external action is derived. This motive can never be a regard to the honesty of the action, for it is a plain fallacy to say that a virtuous motive is requisite to render an action honest, and at the same time that a regard to the honesty is the motive of the action. We can never have a regard to the virtue of an action unless the action be antecedently virtuous. No action can be virtuous but so far as it proceeds from a virtuous motive. A virtuous motive therefore must precede the regard to the virtue, and it is impossible that the virtuous motive and the regard to the virtue can be the same. It is requisite then to find some motive to acts of justice and honesty distinct from our regard to the honesty, and in this lies the great difficulty. For should we say that a concern for our private interest or reputation is the legitimate motive to all honest actions, it would follow that wherever that concern ceases honesty can no longer have place. But it is certain that self-love when it acts at its liberty instead of engaging us to honest actions is the source of all injustice and violence, nor can a man ever correct those vices without correcting and restraining the natural movements of that appetite. But should it be affirmed that the reason or motive of such actions is the regard to public interest, to which nothing is more contrary than examples of injustice and dishonesty, should this be said I would propose the three following considerations as worthy of our attention. First, public interest is not naturally attached to the observation of the rules of justice, but is only connected with it after an artificial convention for the establishment of these rules, as shall be shown more at large hereafter. Secondly, if we suppose that the loan was secret and that it is necessary for the interest of the person that the money be restored in the same manner as when the lender would conceal his riches, in that case the example ceases, and the public is no longer interested in the actions of the borrower, though I suppose there is no moralist who will affirm that the duty and obligation ceases. Thirdly, experience sufficiently proves that men in the ordinary conduct of life look not so far as the public interest when they pay their creditors, perform their promises, and abstain from theft and robbery and injustice of every kind. That is a motive too remote and too sublime to affect the generality of mankind and operate with any force and actions so contrary to private interest as are frequently those of justice and common honesty. In general, it may be affirmed that there is no such passion in human minds as the love of mankind merely as such, independent of personal qualities, of services, or of relation to our self. It is true, there is no human and indeed no sensible creature whose happiness or misery does not in some measure affect us when brought near to us and represented in lively colors. But this proceeds merely from sympathy and is no proof of such an universal affection to mankind since this concern extends itself beyond our own species. An affection betwixt the sexes is a passion evidently implanted in human nature, and this passion not only appears in its peculiar symptoms but also in inflaming every other principle of affection and raising a stronger love from beauty, wit, kindness than what would otherwise flow from them. Were there an universal love among all human creatures, it would appear after the same manner. Any degree of a good quality would cause a stronger affection than the same degree of a bad quality would cause hatred contrary to what we find by experience. Men's tempers are different and some have a propensity to the tender and others to the rougher affections. But in the main we may affirm that man in general or human nature is nothing but the object both of love and hatred and requires some other cause which by a double relation of impressions and ideas may excite these passions. In vain would we endeavor to elude this hypothesis. There are no phenomena that point out any such kind affection to men independent of their merit and every other circumstance. We love company in general, but it is as we love any other amusement. An Englishman in Italy is a friend, a European in China, and perhaps a man would be beloved as such where we to meet him in the moon. But this proceeds only from the relation to ourselves which in these cases gathers force by being confined to a few persons. If public benevolence, therefore, or a regard to the interests of mankind cannot be the original motive to justice, much less can private benevolence or a regard to the interests of the party concerned be this motive. For what if he be my enemy and has given me just cause to hate him? What if he be a vicious man and deserves the hatred of all mankind? What if he be a miser and can make no use of what I would deprive him of? What if he be a profligate debauchee and would rather receive harm than benefit from large possessions? What if I be in necessity and have urgent motives to acquire something to my family? In all these cases the original motive to justice would fail, and consequently the justice itself and along with it all property, right and obligation. A rich man lies under a moral obligation to communicate to those in necessity a share of his superfluities. Where private benevolence the original motive to justice a man would not be obliged to leave others in the possession of more than he is obliged to give them. At least the difference would be very inconsiderable. Men generally fix their affections more on what they are possessed of than on what they never enjoyed. For this reason it would be greater cruelty to dispossess a man of anything than not to give it to him, but who will assert that this is the only foundation of justice. Besides we must consider that the chief reason why men attach themselves so much to their possessions is that they consider them as their property and as secured to them inviolably by the laws of society. But this is a secondary consideration and dependent on the preceding notions of justice and property. A man's property is supposed to be fenced against every mortal in every possible case, but private benevolence is and ought to be weaker in some persons than in others, and in many or indeed in most persons must absolutely fail. Private benevolence therefore is not the original motive of justice. From all this it follows that we have no real or universal motive for observing the laws of equity but the very equity and merit of that observance, and as no action can be equitable or meritorious where it cannot arise from some separate motive, there is here an evident sophistry and reasoning in a circle. Unless therefore we will allow that nature has established a sophistry and rendered it necessary and unavoidable, we must allow that the sense of justice and injustice is not derived from nature but arises artificially, though necessarily, from education and human conventions. I shall add as a corollary to this reasoning that since no action can be laudable or blamable without some motives or impelling passions distinct from the sense of morals, these distinct passions must have a great influence on that sense. It is according to their general force in human nature that we blame or praise. In judging of the beauty of animal bodies, we always carry in our eye the economy of a certain species and where the limbs and features observe that proportion which is common to the species, we pronounce them handsome and beautiful. In like manner, we always consider the natural and usual force of the passions when we determine concerning vice and virtue, and if the passions depart very much from the common measures on either side, they are always disapproved as vicious. A man naturally loves his children better than his nephews, his nephews better than his cousins, his cousins better than strangers, where everything else is equal, hence arise our common measures of duty in preferring the one to the other. Our sense of duty always follows the common and natural course of our passions. To avoid giving offence, I must here observe that when I deny justice to be a natural virtue, I make use of the word natural only as opposed to artificial. In another sense of the word, as no principle of the human mind is more natural than a sense of virtue, so no virtue is more natural than justice. Mankind is an inventive species, and where an invention is obvious and absolutely necessary, it may as properly be said to be natural as anything that proceeds immediately from original principles without the intervention of thought or reflection. Though the rules of justice be artificial, they are not arbitrary, nor is the expression improper to call them laws of nature, if by natural we understand what is common to any species or even if we can find it to mean what is inseparable from the species.


I'm Rick, be safe kids. And hey, this is a really good mask. They have not endorsed this. This is just one that I got on set on something I was doing and I've loved it. This is it. Where are your fricking masks? Is that a N95? It is an N95. Yeah. Nice. Anyways, today, we're, go ahead. Go ahead. The reason we are like this, I'm up in the mountains of Los Angeles was called basically like Airhead Big Bear. We're just me and Stefan Lelander chilling up here for a few days, but a new trailer dropped. And it's for a new topsy-panoo and tarred Raj film. It's called Loop Lapita, Loop Lapita, right? Close enough, yeah. It's a remake or an adaption of Run, what was it? Run Lulu Run. Bob Milkabog? No. It's originally a German film, Run Lulu Something, I think, but it is apparently an original. I've never seen it. I just know it's a remake or an adaption of that film. I think it was from the 90s, actually, that film. But topsy-panoo, one of our favorite actresses.  So really excited about this. I don't know what this office is, but I don't really want to know. So I guess we'll just be surprised here. Yeah, let's just jump in. All right, three, two, one. Wow, the cinematography. Now I just wanted to show you my feet. Ow, ow, ow, ow. It's a nice bag. All it takes is a day. I don't know about life, but your face has definitely changed. Sophie, we're so fucked. My job will be very simple. Take this bag to that address. You'll get a lot of Pedro. He's given you a bag. There will be 50. 50 what? 100,000. 500,000. 500,000. 50,000. 50,000. 50,000. So you're betting? And you lost all the money, right? I'm dead, Savit. You'll cut me, Savit. If you know what I mean.  Don't be Japanat. Who's Karma? Hands up. Satya. Satya. Why are what we doing? CirlaaS. Cirl Suzanne. Satya are you okay? Satya are you okay? I have to say one this time. I have tomorrow. His life is a loop, my son. Just think. Tomorrow morning, I'm going to get your lottery. And you're going to be off today. It's going to be a bit sad. Next, does the save release date? Hold on. Does the save release date? No, I just said watch on any device. February 4th. February 4th, I said just saw in the description. Okay, cool. What a well cut trailer. Like, when we talk about life. That is so interesting. When we talk about like, how nowadays, you know, making trailers is an art form. Because what they do, they know how to cut everything. So it kind of seamlessly goes together and keeps you intrigued. But also doing it to music. And this looks like a very stylized kind of filming in the way they do it. And it also looks like a very interesting story. Almost like Groundhog Day-ish is a vibe I'm getting of like, they're going to have to keep repeating until they get it right, I guess. I don't know. It looks very original. It looks quirky. It looks fun. It looks very original. It looks like it has that combination of both really ridiculous funny moments as well as some poignant deep moments. It looks very, very unique. Looks like it has a good action as well as sound. There was some great sound moments in there when he shot them. A couple of the punches were really, really nice. And obviously we've never seen the original of whatever this film is. So obviously we don't know what's going on. Probably won't see it before this comes out. So if you guys obviously know, do not tell us what's going to happen in this film, please. But Loop, which makes me think it's one of those kind of things that kind of happens over and over again, right? It's the name. Right, right, right. Loop beat that. When a boyfriend loses a mobster's cash, Savvy races against the clock to save the day. If only she can break out of the curious cycle of dead ends is a synopsis. Yeah, I'm intrigued. I'm very intrigued. Oh, hold on. Let's see here. Okay. So the director is the writer of In Grazy Medium, which we saw in Mumbai with no subtitles. No, we did. We had our personal translator in Drani. I'm between the both of us going, okay, well, he said this. Okay, well, he said this. And we never reviewed it because we didn't feel like we had barely seen the film to be able to. Right. Just so you guys know. But he is apparently, oh, he's also the writer. Oh, sorry. I'm looking at the, I'm looking at the writer, which is why it came up. Same writer as that film. The director is, his name is Akash Bhaitia. Yeah, I don't, we don't know. That's the director. So it's the same writer. Same writer as that film. You know, it had, it had some, like some of the quirky fun vibes of Ludo a little bit. Had a little splash of that in there. Yeah. But so original and you're absolutely right. The cinematography, the cutting of this trailer, the sound, it looks, it just looked, I'm intrigued. I really would like to watch it. And the acting looked great. Of course, Topsy, obviously, we, you know, we'd love Topsy up in there. We think she's a phenomenal, phenomenal actress and she always, to my knowledge, does unique work in terms of she likes to pick more interesting projects most of the time than the typical Bollywood style of, you know, just making a lot of money. She likes to pick unique projects. And I like that or projects that will challenge her or different from stuff she's already tried. Yeah. And there's other guy, I don't think we've never seen him anything, but he looks all that guy in the beginning with the beard looked, I thought it was B.J. Sadapathy for a second, but it's like this old guy with a beard, he looks like he's like a theater trained classical actor, honestly. Yeah. He did. I thought I recognized him at first and I was taking a little bit of a closer look at his face. Okay. It looks like his name is Dub. God, I can't pronounce that. But he was in Lutera. He was a Trent. He was a friend. He was in Lutera. Varun's friend. And then apparently he was in Devdie's Chuni. I don't remember any of these parts, but those are obviously a Black Friday he was in apparently. Wow. McBool, he was apparently monsoon wedding. Wow. The brief that we've absolutely seen him multiple times. Yeah. I'm wondering if he's been more of like a supporting actor, but he looks strong. He probably looks different. He probably looks different in this film. And you know her choices, you mentioned Topsy's choices, she reminds me a lot in the difference of her style and the direction she likes to go. Her choices kind of remind me of a Ushman. Yeah. Yeah. Yeah. Very, very kind of similar. Yeah. Outside the box. Not predictable. Yeah, totally agree. A Ushman, I feel like he has his own genre now and it's something that always has a message. And that's like his stick. He wants to have a message in his film. But he's a phenomenal actress and he always sees new films. But Quirky, because like Amer Khan always has a message, but he doesn't go to Rome of the Quirky.  I'm quite intrigued. I think everything about that trailer is really good from the music to the cinematography, to the acting, to the direction of everything. It's a well, well cut trailer. Looks like it comes on February 4th. God, February is filling up. And unfortunately it looks like all this stuff is coming to OTT again. So we're getting OTT releases and not theatrical, which makes me wonder about the Fafa film that was supposed to come at the beginning of February. Remember that unique one? Yeah. It's either going to go to OTT or they're going to push it to March because I think it's pretty obvious that we've got to, I think we'll see the peak globally of Amer Khan here in the next two weeks or so. And then it won't start to taper until beginning of February. And I don't think anybody wants to risk their box office until this is cleared out. So, you know, like, like I have felt the whole time, we'll see numbers finally cascading down by like Valentine's Day. So the first two weeks of February, I think any film there is going to go OTT or push to March. Yeah. But this one looks quite, quite good. Let us know what you thought about it. And if you've seen the original, do not spoil it for anybody. Please. Yeah, please, please. Because obviously if this is an adaption or a remake, a lot of you might know the ending, but do not share it, please. It's down below.


Well students, this is the second lecture on our NPTEL course on numerical analysis. In this lecture, we will recall the Taylor's theorem and how it can be used in approximating a function in a small neighborhood of a point. As the name suggests, this concept was developed by Brooke Taylor as a part of his work on astronomical refractions, but his work was not noticed by mathematicians for a longer time until J. L. Legrangi found it and commented that it is a very fundamental work in differential calculus. Let us start our discussion with the definition of Taylor's polynomial for a function at a point. Let us assume that the function f is n times continuously differentiable at a given point say A. Then the Taylor polynomial of degree n for the function f at the point A is denoted by T n and is defined as T n of x is equal to f of A plus f dash of A into x minus A plus f double dash of A divided by 2 factorial into x minus A the whole square plus f triple dash of A divided by 3 factorial into x minus A the whole cube and it goes up to f nth derivative of A divided by n factorial into x minus A to the power of n. So, that is what is written here and this is called the Taylor polynomial. Let us try to have a graphical idea about what is happening in this polynomial. Let us take this function f of x equal to e power x and we all know that is a very nice function. It is a real analytic function. In this graph the blue solid line is the graph of the function e power x and let us take n is equal to 0 and A is equal to 0. That is we want the polynomial of degree 0 for the function f of x equal to e power x around the point A equal to 0 obviously it will be the constant and you can see what is that constant by putting n equal to 0 in the Taylor polynomial. You can see that T naught of x is equal to 1. This red line is the graphical representation of this 0th degree polynomial T naught of x equal to 1. You can see that the polynomial agrees with the function only at the point A equal to 0. Well, let us increase the degree of the Taylor polynomial and see what is going to happen. Let us take n is equal to 1 that is we want a linear polynomial around the point A equal to 0. Now, it is a straight line given by 1 plus x and its graph is shown in the red line in this graph and as usual blue line is the graph of the function e power x. Now, you can see that well the polynomial agrees exactly with the function value at x equal to 0 and also at least graphically we can see that this polynomial is approximating the exact function in a rather very small interval around 0 right. What it means that suppose I want to find the value of the function e power x say at some point here then instead of taking the value from the exponential function if I take the value from this polynomial then the value seems to be pretty close to the exact value at least graphically. Rather than I go somewhere here and try to find the value of say x is equal to minus 0.5 then if I use the polynomial its value may be very much different from the exact value of the function at the point x equal to minus 0.5 right ok. So, to summarize from the 0th degree polynomial when we came to the linear polynomial we see that we got a small interval in which we may have a better approximation to our function through the Taylor polynomial. Let us increase the degree by one more and see what is happening. For n is equal to 2 now and as usual a equal to 0 the graph of the quadratic Taylor polynomial is shown in this red color and blue line is as usual y is equal to f of x. Now, you can see at least graphically that the Taylor polynomial is approximating the exact function in a little bigger neighborhood of 0 then what we got in the linear case right. So, that means to say that now if I have some point say somewhere here then if I compute the value from the polynomial and the value from the original function f of x equal to e power x the difference may be very small that is what I meant by saying that the polynomial is approximating the function pretty well in a rather little bigger neighborhood than what we got in the linear case. With that idea let me go on with the cubic Taylor polynomial you can see that the approximation is rather good in a bigger neighborhood. Now, I can say that the Taylor polynomial is approximating the function rather in a bigger interval say roughly from minus 0.6 to plus 0.6. Similarly, if I go for the fourth degree polynomial it is much more better and so on. So, in this nice function you can see that by increasing the degree of the Taylor polynomial we are getting better approximation of the function by the polynomial in a bigger and bigger neighborhood of the point A which is taken in this particular example as 0. So, this is the purpose of Taylor polynomial. Now, the question is suppose I want to approximate the function f of x by its Taylor polynomial of degree say n. Now, the question is what is the error involved in it? It means what is this number f of x minus t n of x for any x in a small neighborhood of the point A say A minus delta to A plus delta right something like A is equal to 0 in our case. So, I want to approximate my function f by the Taylor polynomial in a small neighborhood if I do so how this quantity will look like that is the question. So, for this we can go to the Taylor's theorem and see how this error will at least look like. Let f be a n plus 1 times continuously differentiable function on a open interval containing both the points A and the point x at which we want to find the approximate value of the function f. Now, the Taylor's theorem says that you can find a xi between the points A and x such that you can write f of x is equal to the Taylor polynomial plus the error. Remember we were interested in understanding how this quantity will look like.  It means you can represent any sufficiently smooth function by the Taylor's polynomial and whatever you lose as an approximation that can be precisely quantified if you know this xi that looks really too much to expect therefore, xi has to be something which is unknown that is what is also coming from this theorem. If you go through the proof of this theorem you will see that this xi is coming through applying Rolls theorem several times. If you go back to Rolls theorem and see the proof of the Rolls theorem is not constructive it is something conceptual therefore, you just cannot quantify this xi anywhere in this formula. As we have already told T n is the Taylor polynomial of degree n and the second term that is the error term which we will also call as truncation error I will precisely define it later, but in calculus we call it as reminder term. Well I will not prove this theorem however, we have given the proof of this theorem in our notes interested students can go through it. In fact, I will suggest you to go through the proof of this theorem because the idea involved in this theorem is also used in some sense in getting the proof of the error in polynomial interpolations. Therefore, this proof is definitely very helpful for us and it is also important for us to at least once understand in our life. So, that we know how such important theorems are proved. Let us take an example the reminder theorem of the Taylor's formula for the function f of x equal to e power x around the point a equal to 0 is given by this why it is so because you have f n plus 1 of some xi divided by n plus 1 factorial into x minus a to the power of n plus 1. That is the formula here we have taken n is equal to 4 and a equal to 0 where f of x equal to e power x therefore, the reminder formula is e to the power of xi divided by n plus 1 which is 5 factorial into x minus 0 to the power of 5. So, that is how it is coming. What is xi? You should always remember to write what is xi? xi is some unknown lying between x the point at which we want to find the approximate value of the function f and the point 0 which is kept as the point in the Taylor's formula. By the way the formula given in the Taylor's theorem that is this formula is called the Taylor's formula in general. So, the Taylor's theorem says that e power x can be written as the fourth degree polynomial of e power x that is the Taylor polynomial of e power x plus the corresponding reminder term. If you recall the Taylor polynomial of degree 4 for the function e power x is given by this and the reminder term we have already seen that is this. So, these two combined will give you the value of the exponential function at a point x. Unfortunately this term although we know the expression, but its precise value is not known for any given x because the xi is not known to us. So, that is the main problem of this approximation. Just to have a feeling let us take x is equal to 1, the fourth degree polynomial at the point 1 gives 2.7083 and so on whereas, the exact value is something like 2.7183 so on. So, therefore, you can precisely get the error involved in the Taylor polynomial when compared to the exact polynomial in this particular case and that is around 0.01 right. Of course, this is precisely known because we have fixed our x here as 1. In general for a given x we do not know how it will look. If you precisely give x then possibly you can compute provided you know how to compute the exponential function, but as a general case this is not known to us. Then you may ask then what is the point of having this reminder term well we have at least some idea of how that error will look like and in certain cases you can also find an estimate for this error. What is mean by estimate of the error? Let me just give you a idea of it in this particular example. Assume that I want to find approximate value of the function e power x at some point x less than 1. Whatever x I will choose that will be something less than 1. If that is so then what you can do is you take the modulus on both sides of this reminder and that is going to be modulus e psi divided by 5 factorial into x to the power of 5. Now, what I meant by estimating this error? It means I want to find a fixed number. I do not want the upper bound to involve anything which is unknown or something which is a variable.  Therefore, its maximum is actually achieved at x equal to 1 right. Therefore, for any given x whatever may be the value of x you have some xi which unfortunately I do not know, but however what I can say is whatever may be that xi is it will be surely less than or equal to e that is what I will say and then divided by 5 factorial and again x is going to be something less than 1, but here I can say that x to the power of 5 is surely going to be less than 1 because I know precisely how I am going to choose my x. Therefore, with all these ideas I can say roughly that this quantity whatever it is which is we do not know will be surely less than or equal to e to the power of 1 divided by 5 factorial that is approximately 0.0227 ok. So, that is what is finally, we meant by estimate. So, whenever we give a function and ask you to write a Taylor polynomial of degree something some n we will give and also we will give some value for a and then ask you to write the Taylor polynomial for that function then you know how to write because you have the formula and you can also write the reminder term for that and if we want to estimate it then somehow you have to get a upper bound for the reminder quantity which does not involve any unknown or any variable it should be a fixed number that is what we meant by estimates. Well, as I told you at the Taylor's theorem the reminder term is called the truncation error involved in the Taylor polynomial when compared to the exact function often we use this word truncation error in our course. Let us see how Taylor's theorem is used generally you are given a real number a and you are also given a real number h. Let us try to visualize it you are given a and you are given a small number h greater than 0 or even less than 0 if it is greater than 0 a plus h will come this side if it is less than 0 a plus h will come this side either it is greater than 0 or less than 0 anything is ok. And now we are also given an integer n and all these informations are given to us practically this is too much to expect. However, as far as this approximation procedure is concerned we have to know all these information that is f of a f dash of a and so on up to f n of a. Once you know all these informations then the question is what will be the value of f at the point a plus h that is you are given information about what is f of a and similarly what is f dash of a and so on up till f n of a and you now want to know what is the value at a plus h say. Suppose this is the function you know all the informations like this at this point only you do not know at any other point now I want to know what is the value of f of a plus that is precisely what your Taylor's theorem is going to tell you now. The Taylor's theorem says that I can write f of a plus h as this is the Taylor's polynomial. So, if I do not write the truncation error here if I do not write that then I should put approximately equal to. So, you should remember this notation is very important whenever you are writing this this is the exact value that you want to compute unfortunately we do not know this. So, we are going to find an approximate value of that by evaluating the value of the corresponding Taylor's polynomial at the point a plus h that turns out to be this. Remember if you want to find the Taylor's polynomial at x it involves x minus a right. So, now our x is a plus h therefore, a plus h minus a is h that is why h is sitting here then x minus a the whole square therefore, this and so on that is why h is sitting here ok. So, this is the way we will use Taylor's theorem in approximating values of a function by the value of the corresponding Taylor's polynomial and the error involved is the truncation error which is also called the reminder. So, as I told you how to estimate the truncation error let us again assume that f is n plus 1 times continuously differentiable and assume that you are working in a closed and bounded interval say I do not want to use a comma b because a is already reserved for the point around which we want to approximate the function. Therefore, I just used alpha comma beta.  Therefore, you can find a constant such that modulus of f n plus 1 of xi is less than or equal to m n plus 1 that is the constant I am assuming then what you can do recall this is the general formula for reminder term which also is called truncation error. Now you see this can be dominated by m n plus 1 remember by estimate we mean to have an upper bound which is just a number which does not involve any unknown or any variable here xi is a unknown and x is a variable of course, if you change x the xi also will vary right. So, we have to somehow eliminate these two members a is not a variable or unknown it is known to us right. So, therefore, we have to somehow eliminate these two we can eliminate xi by dominating this term by the constant m and you can dominate x by beta because whatever is the value of x it is going to lie between alpha and beta therefore, x will utmost take the value beta. Therefore, we will replace this by beta and this entire term we will replace by m right. So, that will give us an estimate of the truncation error and that is given by m n plus 1 divided by n plus 1 factorial into this one and that further I will eliminate by putting beta instead of x right that is the way we can get the truncation error estimate. Now, let us pause on to what is mean by Taylor's series you are given a C infinity function f then the power series given by this around a point a around which your function is a C infinity function is called the Taylor's series. Now, the question is if the series converges then what is the limit of this series well under certain conditions on f we can say that the limit of this series is equal to f of x well these conditions are listed in the form of a theorem in our notes, but just to keep our discussion simple let us assume that f is a real analytic function then of course by definition f of x can be represented by the Taylor series right. Of course, we know this from our calculus course let us take an example f of x is equal to cos x we know that the Taylor's series at a point x equal to a for the function cos x is given by this right. Now, let me take in particular a is equal to 0 then this series reduces to 1 minus x square by 2 and so on remember all the terms with sign will vanish that is why we do not see the odd powers here only even powers survive and therefore, we can write the series for cos like this right ok. So, why am I telling this we all know this now coming back to Taylor formula at a equal to 0 n equal to 4 for the cos function can be taken like this ok. Now, I am putting approximately equal to because I am not writing the truncation error or the reminder term here. Therefore, I have to put the approximately equal to symbol here and what I am losing in this approximation is precisely given by this expression. However, its value is not known to us precisely, but the expression is known to us like this where the psi which is the unknown lies between 0 and x ok. So, here what I would like to highlight is generally you may write the reminder term just going by the Taylor's theorem may be written with x to the power of 5 term right here, but what we are taking as a reminder term is x power 6 term why we are doing it because that is the immediate next term in your series right and therefore, that dominates your truncated part. What is that you are truncating x to the power of 6 divided by something plus x to the power of 8 divided by something and so on. So, what we are truncating is this part and the leading term of that part is x power 6. Therefore, we actually has to take this as the reminder term and not with x to the power of 5. This one has to remember why we are doing this well when we do the order of convergence at that time we will come and revisit this problem and tell why we prefer to take this term as the reminder term and not the one with x power 5 which actually your Taylor's theorem may suggest ok. So, in that way your Taylor approximation for cos function may be written like this where the index goes from 0 to m with 2 into k here and the reminder is immediately taken at the m plus 1 stage ok. So, this is something that you have to keep in mind when you are writing the reminder term for cos x similarly for sin x also ok. With this our discussion or Taylor approximation is over we will meet you in the next class. Thank you.