Now, let us begin our discussion of the experimental skills. Here, we will cover the following points shown on the slide. First, we will illustrate what is the meaning of the scientific method with an example. Then we will see variables and errors in experiments. Then what is meant by control, precision and accuracy. And finally, we will mention in passing how to fit a straight line to measure data. Let us look at the scientific method. What does it involve? One can broadly identify four steps in this method. First is the observation of phenomena. Then based on the observation and our thinking, we come up with a hypothesis about some happening in the world around us. It can… The hypothesis is some sort of an imaginative explanation for things which we observe. However, it is not verified to be correct. So, hypothesis is a sort of statement of explanation for the phenomena which is not yet verified to be correct. So, it is mostly based on intuition. Then comes the important step of verification of the hypothesis. So, here broadly there are three steps, experiment, observation and inference. If one wants, one can club this entire thing, step of verification, all the three steps of experiment, observation and inference in just one word that is experiment. And after the experiment and the inference drawn from it, you make a generalization to all situations which may be similar to that considered in the experiment. Now, let us look at the various elements of this method in little bit more detail. See observation. Observation is not a passive acquisition of sensory information, but a critical purposeful process needing a high level of awareness. This high level of awareness is very, very important for being a good observer because when you say that you should be observant, what does it mean? What is it that you are supposed to observe? So, when you ask a student, if the student is doing some chemical reaction and then you tell him you be observant and see how variables in the reaction are changing. The point is you are not telling him what are all the variables that the student should observe. It is not possible to tell also because after all you are doing a new experiment, it is your duty to be aware as much as possible to identify as many variables which may be affecting the experiment. So, it is in this sense a very high level of awareness is what is expected if you want to be a good observer because you cannot tell beforehand what are the things that you should observe. So, for example, you are doing a chemical reaction and you are concentrating on measuring the temperature, but at the same time the reaction is been done in a environment where there is light because otherwise you cannot see what is going on. It is possible that lighting may have an effect. So, like this it all depends on a person how much aware he is depends on the awareness whether you are able to observe the presence of a variable which is affecting the reaction. So, there is a reason why we choose to observe a specific portion of the whole domain of possible objects of observation. So, it is not as though something that is very evident to your eye or something that can be heard very clearly alone is chosen for observation. You are aware of several things that are happening around you, but then even though you are aware of all the things you choose only a part of those things that you are aware of for critical enquiry and careful observation. Hypothesis Hypothesis is an imaginative preconception or an inspired guess about some particularly interesting aspect of the world. Every discovery begins as a hypothesis. Now, hypothesis is based on intuition it may or may not be correct. We will illustrate this with examples. Now, what is the experiment? Experiment is the act undertaken to verify a hypothesis. It discriminates between possibilities and gives direction for further thought. This is another role of experiment. It discriminates between possibilities. It is a happening device to apprehend the truth. So, it is a set of conditions that you ensure in order to observe a particular thing. So, for example, if you want to detect the wave nature of matter, then you have to devise conditions, special conditions so that this wave nature will become evident. If you want to detect the particle nature of light, you have to devise special conditions to detect that particular aspect. For knowing the wave nature of light, you may not have to devise special conditions because already the conditions which are existing in the environment may be sufficient for you to observe easily the wave nature of light. So, here you are not devising special conditions. It is a method of discovering causal interconnections. So, what is the experiment? Another way of looking at experiment is it is a method of discovering causal interconnections. Hypothesis. Devising a hypothesis that can be tested by practical experiments is an art. So, every hypothesis may not be possible to test every hypothesis in a scientific way or to establish whether it is true or not. For example, the hypothesis is all manner of model, it is not something that you can test. It is not possible to scientifically examine this hypothesis. All men are model. Then how do you frame a hypothesis? That is also important. So, there is a method of framing the hypothesis, it is called null hypothesis. That is hypothesis stated in negative terms. I will leave it as an assignment to you to see this point in detail. What is the meaning of a null hypothesis or framing a hypothesis in negative terms? In a very simple way, I could put it as follows. Once you make a hypothesis, some conditions A result in effect B. So, A implies B that is your hypothesis suppose. Now an equivalent statement of the same hypothesis would be if there is no B there is no A. A implies B is equivalent to saying B bar implies A bar. B bar means not B. Meaning you are making a hypothesis. Some variable A results in an effect B. It is same as saying if you do not see the effect B then it means A is not present. Not B implies not A or no B implies no A. Now this statement and any of the statements are equivalent but when you frame the statement in this form not B implies not A, no B implies no A that is if you do not find the effect B then you can be sure that the condition A is absent. This form of framing the hypothesis is called a null hypothesis. Many people say that to test hypothesis it is better if you test its null form of a hypothesis. So more about this I leave it as an assignment because framing of hypothesis itself is important and there are several details about this. Now we will consider an example to illustrate this particular aspect of the scientific method. In the west one observes a high level of prevalence of obesity. So there is a large number of people who are obese. In fact about 50% of the population is overweight, this is an observation. So naturally people are concerned about it. So some scientist working on the issue of obesity and removal of obesity they hypothesized that the overweight people are overweight because they have a higher level of hunger than the average weight people. Because they have a higher level of hunger they eat more and therefore they become overweight. So the fact that people are overweight is one observation. The fact that overweight people are seen to eat more is also an observation and then you are trying to frame a hypothesis by intuition. What is the cause of overeating? What is the cause of overeating? So the cause of overeating is more hunger. Now some scientist came up and said look I do not accept this hypothesis that the hunger the higher level of hunger is the cause of overeating. I want to test whether it is true. So he suggested an experiment or rather he carried out an experiment. Now his experiment was as follows. He took 100 overweight people and 100 average weight people. This group of 200 people was divided into 2 groups of 100 people each. Each group of 100 people had 50 overweight people and 50 average weight people. Now each of the 2 groups were given sandwiches to eat. So one group of 100 was put in one room and each person was given one sandwich to eat. The other group of 100 people was put in another room and they were given 3 sandwiches to eat and people in both the rooms were told that if they do not want they need not eat or if one sandwich is not sufficient to you know satisfy their hunger they can go and fetch another or more sandwiches from room which was kept about 50 meters away. So they can bring sandwiches from there and eat if one sandwich is not sufficient. Now after carrying out this experiment they measured what is the average number of sandwiches eaten by the overweight people and the average weight people. Now very interestingly they found that on average the average weight people ate only one sandwich in both the rooms. So in the room where I am sorry I correct myself the average weight people ate 2 sandwiches in both the rooms which means in room number 1 where only one sandwich was provided most of them fetched one more sandwich. In the second room they left out one sandwich they did not eat all the 3 sandwiches. However overweight people the behavior was different in room number 1 most of them ate only one sandwich whereas in room number 2 most of them ate all the 3 sandwiches. Now this experiment showed that internal factors such as hunger may not explain this observation because in this case I mean it depends on the experimental conditions sufficient care was taken to see that everyone was sufficiently hungry and so on. There are many other details of the experiment so which we will not go into here. We will assume that conditions were maintained identical in both rooms but what it showed is that for the overweight people the amount they eat depends on the availability of food which is an external condition not an internal condition to the person. So the eating habits of overweight people are stimulated by external conditions in the environment more specifically availability of food. For the average weight people however there is an internal control mechanism independent of the amount of food available they will eat some amount that is controlled. So this experiment therefore showed that you cannot assume hunger as the cause or higher level of hunger as the cause for overeating. Of course there have been further experiments and so on whatever I have said is a simplified form of the situation but nevertheless it illustrates the importance of scientific approach. So what is the scientific approach? Scientific approach is you must verify the hypothesis nothing is accepted until you test and you must devise an experiment. So that is also important in science how do you think of an experiment to test your hypothesis and testing of the hypothesis does not mean that you should prove it even if you disprove a statement right even if you disprove a statement it is significant. Now the lack of scientific method however or scientific approach is very prevalent in society. Now before I give an example of a non-scientific method right I want to emphasize the fact that it is possible that not every hypothesis can be tested we have already said this. So there are many things which you may not be able to test by the scientific method devise an experiment and test and you cannot say that what is not possible to test by a scientific approach is not true that is not the kind of thing the statement that we are making but for our purpose of research right in a wider philosophical sense it may be true that everything cannot the truth all aspects of truth cannot be scientifically tested it may be true here we are only talking about research in science and engineering or humanities even humanities where you can test things scientifically but when you say testing scientifically you should be clear what it means. So what it means is in a scientific method two important elements are there a hypothesis and an experiment so both things are important an example of not using verification it happens very often in life but a significant example of this is the belief of I forget whether it was Aristotle or Socrates one of these two. So this philosopher believed that women had less teeth than men now this is a hypothesis it was based on intuition and once since the philosopher was well established and well known people also believed this statement and this happened for several hundred years. Now no one attempted to ask a woman to open the mouth and simply test check how many teeth she has this is very interesting why because the person who made the statement was very well established and regarded as a good thinker. So we must be aware that very often we might use non-scientific approach and we may have beliefs which are not tested we must always try to test the beliefs as far as possible. So this is an example of thinking based on only intuition very often all our hypothesis are based on intuition without intuition you cannot work. So we are not saying that you know intuition is bad intuition is very very important in fact only intuition can give you hypothesis but intuition is not sufficient in scientific working in working by scientific method intuition alone is not sufficient you must verify the verification or experiment is very very important. So now let us look at some other aspects of experiment conducting experiments types of variables independent variable stimulus or input or cause all these are synonymous terms. So this is the condition manipulated to determine its effect on an observed phenomenon then some variables can be dependent variables. So dependent variables or response or output or effect so this is the condition that appears disappears or varies as an independent variable is introduced removed or varied very important point in some cases it may be difficult to decide which of a pair of variables is the cause and which is the effect. So just because you see there is a relationship between two variables two variables are correlated does not mean you will know which one is the independent variable and which is the dependent variable then apart from independent and dependent variables you also have another type of variables they are called confounded variables confounded variable is one whose effect cannot be separated from the supposed independent variable. Now we will we can illustrate this with example suppose you are trying to find out if men and women have any specific preference for colors you want to now this is a hypothesis that you want to test whether there is a specific preference. Now you think you take a group of men and group of women and then you do some experiments with them and you find some difference and you try to attribute that difference in preferences to the gender but what you are not taking care of is the fact that this women men and women are they have a particular age. So over the years they might have been exposed to particular type of colors this is called past experience you do not know supposing because people are exposed to a particular type of color they like it they may develop a liking for it now this past experience is something it is a confounding confounded variable because its effect you cannot separate out very easily therefore if you can devise an experiment in which you can separate out the effect of the confounding variables or confounded variables then that is something significant. So one must always be aware that there can be dependent variables there can be independent variables and there can be variables whose effect cannot be separated from that of other independent variables and confounded variables. So your experiment should not have confounded variables or you should have some method of eliminating the effects or at least if you cannot eliminate you can at least state in your experiment that this is a confounded variable whose effect we have not been able to detect some other attributes of variables quantitative variable it is something that varies in amount you can assign a number for it categorical variable it varies in kind gender is a categorical variable male or female continuous and discrete variables. So you can have variations of quantities in discrete manner or continuous manner now we have discussed the various types of variable in the experiment let us look at another very important issue any experiment you are making a measurement without a measurement there is no experiment I have emphasized this fact in my introduction that measurement is very very important just qualitative understanding of how when one variable is increased the other variable increases or decreases is not sufficient that is that does not complete the experiment you must measure. So that is where errors coming so two types of errors can be there in measurement random errors. So random error varies and equally likely to be positive or negative it is an error which is equally likely to be positive or negative as against this a systematic error is something which is constant within course throughout the experiment the word constant is to be not to be taken literally in fact if you want you can say non random errors you can say non random errors so random error is one which varies equally likely which is equally likely to be positive or negative that is nature of random error but systematic error it has a certain trend predictable trend okay traditionally the word systematic error has been used so therefore we will retain the error retain this word but you can understand this as a non random error and let us illustrate with example supposing you are measuring the terminal velocity of a ball in viscous medium like glycerin suppose this is the experiment you are doing now let me draw a diagram to show that so this experiment all of us who are in engineering or physics would have done in our first year now this is a jar or a tube and it is filled with glycerin which is the viscous medium and you drop a ball in this medium now the ball initially accelerates downwards because of gravity but ultimately it acquires a constant velocity because as its velocity increases the viscosity produces a frictional force which increases with velocity so for some velocity the downward force because of gravity is exactly balanced by the frictional force or opposing force because of the viscosity of glycerin if you were to drop the ball in vacuum then it would accelerate continuously it will not reach a constant velocity experiment is to measure the terminal velocity of this ball terminal velocity is a constant velocity that it acquires after some time now how will you measure it the common method of measuring is you note for example when it crosses this particular mark with a stop clock so you start a clock then when the ball crosses a different mark you stop so the clock would have come somewhere here some time has elapsed now you measure this distance and measure the time and then the terminal velocity is given by distance by time so these experiment so in this experiment what are the sources of error and how do you classify the errors as random and systematic as shown in this slide random errors can occur in starting and stopping the clock or in estimating the ball location on a scale okay so let us look at the diagram again now you say you are noting the time when the ball crosses this mark now are you noting when the bottom end of the ball is crossing the mark or when the top end of the ball is crossing or is it that you are noting the middle this is one point and it may not be possible for you to precisely note any one of this so there can be some error because you are repeatedly when you do the experiment repeatedly right this something that you are not able to determine precisely and it can be random then starting and stopping of the clock you are looking at the ball and at the same time you have to start the clock so there can be a delay between the two events so you may start the clock before the ball has crossed the particular mark or you may do it a little bit later and this thing cannot be controlled so these are the two sources of random error in this experiment these are examples of two sources of error which are random now what is the systematic error here that can happen so supposing your clock after you start progressively become slow this is an example of a systematic error or when you repeat the experiment what happens is the there is a friction between the ball and the glycerin now this friction causes the glycerin to heat up after all how is the energy dissipated how is the friction the energy dissipated because of friction it is dissipated as heat so the temperature of glycerin will go on rising if you take a very long time to complete your experiment so this is another example of systematic error so like this in any experiment you have either systematic errors or random errors now you should have the ability to distinguish between random and systematic errors and you should also be aware of the fact that these errors are present and what are the sources of these errors because only then you can reduce the errors it is very very important to reduce the errors we will concern example to illustrate why reducing errors is very very important so before we go to that particular example let us see some further facts about errors we normally use the terms accuracy and precision so what is the meaning of accuracy now on the left hand side here you see a point or a circle which can be regarded as a true value of the particular variable being measured and the crosses around it indicate the values obtained in repeated experiments of the same to give you an analogy in fact you can regard I imagine that this is this represents a shooting event where the solid circle is a target and the crosses indicate the places where the bullet strikes okay when a person is trying to shoot at the target in repeated trials now you will say that the marksman is accurate if all these places where the bullet hits at somewhere near the target are very close to the target all of them should be very close so what it means is both systematic and random errors are small so random errors are represented by the differences between the repeated measurements and systematic error is represented by the difference between location of these crosses and the target so here for example when all these crosses are very close to each other but they are located away from the target then you say that the measurement is precise but inaccurate so precision has to do with ability to repeat the value in successive measurements several trials so you may be getting the same value in repeated measurements that doesn't mean it is accurate it means it is precise so accuracy means this value should be very close to the true value so systematic error should not be there precision means random errors are minimum okay but that doesn't mean systematic errors are absent so imprecise and inaccurate both systematic and random errors are present so this is an important point that we should note the difference between precision and accuracy so precision means small random errors accuracy means both systematic and random error should be small now when you write the reading let us see what are we pointing towards the systematic error or the random error now here for example on this slide you show the measurement of a resistance in some experiment and you write the measurement as the measured value as one particular number plus or minus another number so in this case does plus or minus 0.001 ohm represent a systematic or random error what does it represent evidently it is random error because it is plus or minus so it is equally likely to be positive or negative so it is not systematic error so note that in when you report your measurements in this form you are specifying the random error you are not specifying the systematic error now how do you detect and minimize errors this is the next issue random errors can be detected by repeating the experiment so when you repeat the experiment if you are getting different values then you become aware that there are random errors in the experiment however repeated measurements may or may not reveal systematic error this is very important you cannot reduce systematic error merely by repeating the experiment systematic errors are somewhat more difficult to detect and reduce random errors can always be eliminated by repeating the experiment so many times the research scholars are not aware that when they are minimizing error by repeating the experiment they are only removing the random error or let me not use the word remove they are reducing the random error you cannot remove systematic errors by this approach so first of all many times in fact the research scholars are not aware of the errors at all now how do you know whether a research college is aware of errors or not it you can easily see the way the readings are presented in graphical form measurement okay for example if you find that some measurements are reported like this so I show points these are measured points it's a measurement of y as a function of x and then you say the behavior is something like this now clearly the person who has drawn the diagram is not aware of errors because you assume that the measured value is precisely this point right whereas just now we have shown on the slide that supposing you are reporting measurement of resistance you will report it as a value that you have read on the meter plus or minus a quantity plus or minus a quantity that is a random error now I will not discuss in detail how do you decide the random error in a situation right you can look at it I am just initiating the whole discussion about the errors so how do you know the research scholar is aware of the errors well if he shows an error bar if he shows the reading as a range and then this value is the midpoint of this range then you know that yes the research scholar is aware of the errors now why is this bar very important because now you see if I take lowest point here the highest point here because this is also possible and then the lowest point here then I don't seem to discern any specific trend but then you may say this will happen all the time the point is what is this what is the length of this bar this is what is important if the length of this particular bar is small then you can definitely disun some trend but if the length of this bar is long which is what we have seen in our lectures yesterday when we discussed about graphical approach and so on how to represent data in graphical form then you may not be able to discern trend any trend in such a case if you are reporting that you have discern some trend then your conclusion can be in doubt now how to minimize random error you do it by averaging now how does averaging reduce random error let us look at this point also now before we do that let us look at the aspect of control in experiment when do you say an experiment is well controlled so control implies awareness of all the variables in an experiment and the ability to vary them at will so you should be aware of all the variables in the experiment and you should be able to vary all the variables the way you want this enables one to restrain sources of variability in research control also means a standard against which the effect of a particular variable can be compared so you are comparing against a standard the measurement against a standard the effects of all but the independent variable can be eliminated okay effects of all independent variable effects of all variables except for the independent variable now as we have mentioned in one of our earlier lectures the several independent variables can be there and then you will have one effect now how do you find out what is the relation between the different independent variables and the dependent variable so when you want to determine this you must have one independent variable being changed and monitor the effect and at that time the effects of other independent variables should not be there this is one important thing that you should ensure in your experiment this is what is the meaning of control another meaning of control now how do you can how can you ensure that so there are several ways of ensuring depends on the situation so the effects of all but the independent variable can be eliminated by either removing them if you cannot remove them by maintaining them constant or by screening them there is a method called screening you can screen the effect or by counterbalancing counterbalancing means you cancel the effect of one variable with the other variable and there is another approach that is called systematic randomization now all these various methods are very very important I do not have a time to go through them you can note down these words such as screening counterbalancing and systematic randomization and you can try to find out yourself this is an assignment because removing them one can understand maintaining a variable constant this is also clear screening what is the meaning of screening it is clear but how do you do screening right please find out how do you do counterbalancing some practical examples and how do you do systematic randomization we will see some example of how you can remove the effect when you cannot how you can remove the effect of the variable other than independent variable when you cannot remove the variable or you cannot maintain it constant one example we will see now how can there be difficulty in control let us take some examples to illustrate how difficulties arise in control of experiments consider an experiment related to magnetostriction so what is magnetostriction the experiment for measuring magnetostriction is shown on this paper slide so magnetostriction means change in the length of the material because of magnetism change in the length of the material because of magnetism that is magnetostriction you know that when you heat some material it will elongate so that is change in length because of heat like that you can have change in length because of magnetism now supposing you are doing an experiment to detect whether a particular material has this property of magnetostriction so a schematic diagram of the experiment would be like this so you have a solenoid to create the magnetic field and inside this solenoid inside this coil sorry you have a coil to create a magnetic field and then inside that you put the material for which you want to measure the magnetostriction now when you switch on the current magnetic field is created and then you make the measurement of the length from this end to this end and you indeed find for example suppose the length has increased supposing in a particular situation you found the increase in length was say 0.005 percent increase in length because of magnetism now you are happy that you know yes you have detected magnetostriction in the material but unfortunately what you have failed to notice is that when you pass current through the coil the coil gets heated up and because the coil is heated up the material is heated up and because the material is heated up it will expand now are you sure that the expansion that has happened that you have detected or measured is solely because of the magnetism and not because of the temperature so what you have to do you have to measure the coefficient of expansion because of thermal reasons of the material separately and then you should try to see whether the expansion that has occurred here is more than that or you will have to find some method of measuring keeping the temperature constant but this is an example where you can see that when you want to increase the magnetic field the temperature increases okay so these are the kind of difficulties which are there in control another example terminal velocity of a body in viscous liquid like glycerin we have already discussed the apparatus by which you measure the terminal velocity now as we have said the temperature of the glycerin can go on increasing as you increase the number of trials of the experiment so now if you are measuring the terminal velocity as a function of for example diameter of the ball you are doing an experiment to measure terminal velocity as a function of diameter suppose this experiment then you have balls of different diameter and you are going on doing experiment with the different balls and you think that you have seen some trend as the diameter increases terminal velocity increases or decreases whatever you have seen some trend and you think that is because of the viscosity of the medium but the temperature of the medium has also changed so how do you know that the terminal velocity behavior that you have measured is not because of temperature right and because of viscosity alone so here again there is a problem in control then the third example is something that I already mentioned gender differences in color preferences gender is confounded with past experience so past experience confounding variable now in the previous examples the effect of temperature is the confounding variable in magnetostriction experiment and terminal velocity experiment now why should one be bothered about precision why should one try to be as precise as possible so when experiments give results at variance with theoretical ideas new discoveries are made right this is the main reason why you must be bothered about precision let us consider two examples where precision played a role in great discoveries discovery of argon by rayleigh and ramsey this is first example now how was argon discovered now people knew that air contains nitrogen oxygen and traces of water vapor carbon dioxide and so on this is what people had known before argon was discovered now someone did an experiment to see the density of nitrogen that you get in two different to separate methods of getting nitrogen so one approach was you remove the oxygen from air and you try to remove the moisture and so on also since you are not aware of any presence of any other gas so you assume that when you remove oxygen and traces of moisture and so on what is left is nitrogen so you measure the density of the remaining gas let us put it like this now you think it is nitrogen that is your hypothesis another experiment was done where nitrogen was produced from ammonia that approach produces pure nitrogen you are very sure it produces nitrogen it was found that the density of the air that is left when you remove oxygen and moisture and so on was very marginally higher than the density of nitrogen that was produced from ammonia now marginally means how much it was just half percent higher now many times you may say this half percent kind of error can occur in your measurement anyway why bother about it but then when they did the experiment very precisely where this half percent also became significant then and they investigated the cause of this discrepancy they found that you can explain this discrepancy by assuming that there is one percent argon another gas of a particular weight and so on which they called it as argon so one percent of argon was you know detected and that is how the argon was discovered so you can see how precision is very important another example of discovery because of very high precision in experiment discovery of deuterium by dirge and menzel 1931 and ure brick vede and murphy 1932 how did this discovery occur so people produced hydrogen in two different ways and tried to measure the mass you call it as atomic mass one method was chemical determination as shown on this slide and the paper slide the chemical determination method it gave the ratio of the mass of hydrogen that was obtained by this method to mass of oxygen with atomic mass of 16 particular isotope now this measurement was reported as shown here that is 1.00799 plus or minus 0.00002 in a different experiment the mass of hydrogen ratio similar ratio mass of hydrogen to mass of O16 was found to be 1.00778 plus or minus 0.00005 now 1.00799 plus or minus 0.00002 what this means is your numerator can be anywhere between 1.007997 to 1.007801 numerator here here however the numerator can be 1.00773 to 1.00783 now you see that this range is definitely lower than this range even when you include the errors so definitely here the mass of hydrogen was different in this particular experiment than in the chemical determination method now once this difference was detected and then it was hypothesized that you can explain this difference if you assume a very small percentage of hydrogen with atomic mass of 2 instead of 1 that is deuterium so if you assume one part in 5000 of a hydrogen atom with atomic mass of 2 you can explain this discrepancy and that is how with further experiments it was shown that yes this is indeed correct spectrum of hydrogen was taken in a very sensitive experiment and then it was found that you did have lines corresponding to a hydrogen atom of having a mass of two units so it is another example where precision is very important so if your this plus or minus value was much larger you would not have detected any significant difference in the two cases so this where it was very very important to reduce these numbers which means you want your measurements to be made very precise so this is how precision is very very important in our experiments so we can reemphasize what we have said just now with another example significance of results depends on precision if for example the measured resistance at two different temperatures is 2.0025 and 2.0034 we can say that temperature has an effect on the resistance can we say this yes if the error is plus or minus 0.001 ohm no if the error is plus or minus 0.01 ohm why because you can see that if the error is plus or minus 0.001 ohm then your values of resistance at 10 degree centigrade can be 200.024 to 200.026 and at 20 degree centigrade it would be 200.033 to 200.035 now clearly the higher value here the lower value here is significantly more than the upper bound here so clearly at 20 degree centigrade the resistance is different than at 10 degree centigrade so you can say yes temperature has an effect but supposing your error was 0.01 ohm then your values would be 10 degree centigrade 200.015 to 200.035 this is one range that is possible at 10 degree centigrade at 20 degree centigrade it is 200.024 to 200.044 your measurement could happen anywhere in this range now you can see that 200.024 the lower end lies in between 200.015 and 200.035 so supposing these were the actual values then you find that as a function of temperature the resistance is decreasing on the other hand if these were the values it is increasing so you cannot discern any particular trend from here so this is how precision is very very important so if you plot the same thing on a graph I have now shown it in terms of numbers I will leave it as an assignment that the same illustration can be much clearer if you were to plot the things on a graph right your graph would look something like this when the error is small your two readings at 10 degree centigrade and 20 degree centigrade this resistance this is temperature they would be like this so the maximum value here is less than the minimum value there so you can clearly see yes there is an increase this is when the error is 0.001 if the error is 0.01 what would happen is on a graph it would appear like this 10 degree centigrade 20 degree centigrade this is how it would be so now you cannot say you may it may also be that the resistance is not changing at all or it may be that it is decreasing or it may be increasing okay but in this case it is very clear so this is how this actually represents this length represents the precision or the extent of random errors how much precision now we have seen that the precision is important now how much precision we should aim at it depends on the purpose of the experiment if a resistance is to be used as a standard in the range 10 to 20 degree centigrade and the precision required is 0.01 percent an error of 0.01 plus or minus ohm is adequate and it is unnecessary to reduce the error to plus or minus 0.001 okay so in your experiment you should have an idea of how much precision is required so this is another skill that you should have now each of these points can be discussed in great detail with for the examples so you can have several hours of experimental skill right courses on experimental so here i am by just pointing out the important issues you can read up on your own you know what kind of research you are doing the goal of this discussion is to make you aware of the various issues you can you know read up the required material and then understand these issues in depth


This one item that I was supposed to talk about yesterday was the notions of sight, which is important in the material tokens. So I would like to start today with discussing a bit this notion of sight. And then I will talk about the logical aspect of total theory. So I may use the first half hour for discussing the notions of sight. Well, a sight is, let's say C, J, is a small category. C equipped with a gluten-dictapology, let's say J. So I will describe more precisely what this means. Olivia already said something about it yesterday. But I want to stress the fact that what we do is that we have the category of perceived on C, which is a topos. And we somehow want to take a quotient. I'm thinking of the gluten-dictapology as providing some kind of relation so that you can take a quotient. So you have a quotient map, Q, which is a geometric, well, not exactly a geometric, an algebraic morphism of topos. So Q, preserve, put in it, not add in it. It's an algebraic morphism of topos. You may want, I mean, from a geometric point of view, you may want to see this as a subtropical of that, so you reverse the picture. But I want to take up it as a quotient. And the gluten-dictapology is really what you need to take a quotient of the topos from an algebraic point of view. So I would like to discuss these questions of quotient and topology, first maybe with soup lattices. So suppose that L is a soup lattice. And suppose that you have a set of pairs of elements, and you want to take a quotient of L in the category of soup lattices. So a quotient L is a quotient. So has to identify, you want to take the equalizer of all these pairs. So you want to have the QA equal QB for all AB in a universal way. Well, what you do normally, I mean in algebra, is that you construct the congruence of a relation generated by pi. Then you take the quotient, maybe here. So this is a relation for the congruence of a relation generated by pi. But in the category of soup lattices, something very nice happened. Is that you can construct this quotient right away. So let me write L pi, or let me say the definition. An element in X in L is, let's say, pi-sheaf. For all AB in R, A is smaller than 0. In other words, if you call an element pi-sheaf, the element is not able to see the difference between A and B. To see the difference on this side. So in other words, X is a pi-sheaf. If one this is true, this must be true also, and conversely. Now, if you write L pi for a set of X and L, X is a pi-sheaf. But this subset of L is actually closer under a theme. This is really very simple. Let me very quickly sketch the proof. Suppose that you have a family of elements in L. I want to prove that the infimum, let me write X, the infimum, is again in L pi. Well, suppose that for example that A is smaller than X, then of course all I. And since this is a pi-sheaf, and since this is true for all I, and the reverse implication is true. So this is close under an infimum. And since this is close under an infimum, it implies that the inclusion has a left adjoint. If I call the inclusion U, then the inclusion has a left adjoint to U over 3 from L to L pi. This is true from, I can write a formula for U over 3 of something, U over 3 of NMX will be the infimum of over all Y such that X is greater than Y, and Y is in L pi. That's a formula for it.  And the solution to the universal problem of identifying A and B. So if you have a map F with the conditions of F of A, B, or all AB, then you will be able to, I think it has a bigger size, so you can expand it. But what about if the lattice is a frame? Suppose that L is a frame. Suppose that L is a frame, therefore, I mean, it means that the infimum is distributed, right, over super... it's a lattice, but with the extra property that we have this distribution to go down. Okay? And when is it true that the quotient there is a frame? You would like to have the quotient to be a frame now, okay? What I just described is a kind of purely linear theory in the sense that the suplexes are like BN groups, they are randomized like addition, and I just told you how to make a quotient of an AB group by something, right? But now suppose that L is a ring, and you would like to know when is the quotient a ring and when is it the solutions of the problem in the ring aspect. So, but you need that five is... so let's say G and L is the same generator. And now what kind of generator? I mean a suplex generator or a frame generator? Let's say there is a difference between the two. I think frame generator. So I'm just now supposed to... maybe I should be careful. I will take suplex generator. So this is a bit... yes. So suplex generator. And I'm going to say that five is... so it's a stable or something. So the condition now, have the condition add the condition that G is contained in five. So what does it mean? It means that G in G and AB in five implies that G, A, G, B. It is stability under put back, or if you like, intersection in five. Then the same construction give you a frame quotient. A frame quotient. And it solved the problem of taking a quotient of the frame with respect to the given system of relation. So this is a little bit like saying, yeah, that's an idea. And I put in the condition here only for a set of generator because very often that's the way you want to check if something... You don't need to check for every element of five. You just need to check for a set of generator. Again, this is not difficult to prove and it's like you will probably enjoy proving it yourself. So I will not take this pleasure from you. Okay. Now, this pattern is just repeating itself in shift theory. This is exactly the same thing. However, I would like here to make a point about the fact that the pre-sheaf on which is D of P. So if you have a pro-set P and you have the unit that embedding Y, D of P is the pro-set of downward sections of P. And we know that this is the completions of P under Suprema as a suplex. This is the solutions of the linear problem. But it happens that D of P is also a frame. Okay? It's a frame. It's given to us as a frame. But as a frame, it's not to be. This is a frame, but it's not the frame freely generated by P. The frame freely generated by P has another construction. We first add finite meat to P. Okay? And then we complete it by adding Suprema. That's the frame freely generated by P. That's this thing. This is how it works. So this one is not free. The generators, of course, are the YX, okay? D of P is generated by the YX with X and P. But there are the relations. So I would like to discuss the relation between the generator of D of P as a frame. The first relation is that the Supremum of all the generator is the top element. You see, this was true also. What I'm going to write will be true in D of P as a Sup lattice. But I cannot write this relation in the theory of Sup lattice because this element here belongs to the notions of frame. In a Sup lattice, there is a top element, but it has no name. Well, I can say it's the top element, but it's not preserved by morphism of Sup lattice. So if I write these relations, I want to write the relation between using formulas in the theory of frame. And I cannot write this relation in the theory of Sup lattice because this element 1 is not a constant or an operations in the theory of Sup lattice. It's not preserved. So that's the first relation. And the second relation is that these relations are very easy to check, right? I mean, the first just means that P, the top element, is the union. You see, if I use, since I am in D P, I can use union, and this is clear, right? Every element is the similarity from this one, okay? You see, why A meets why B is not a generator in general. I mean, if P was a meek lattice, then why A meets why B would be equal to Y of A meet B? Because the unit are embedding preserve intersection when they exist. But when they don't exist, the intersection between the two is not representable. But however, it would be a suprema of things that are representable. And you would just write it down, okay? So these are the two relations. And it's an easy exercise to check that this is a presentation of D P in the theory of Sup lattice in the sense that if you have a frame A, and if you have a map phi that satisfies these two relations where A is a frame, then there would be the linear extensions which is just obtained by using suprema for S downward sections of D. The diagram is not D but P. Ah, thank you. Yes, right. So you have a map, you extend it linearly, and then you prove that phi prime preserved in the section just from 1 and 2. 1 and 2 is exactly what you need in order to prove that, okay? So in particular, the point of a frame, we call it the point of a frame is a morphogen, is a character. So what are the points of D of D of P? Subsets in P. That satisfies such that two conditions. P is 90, F is 90, and 2, A, B in F implies... Well, I should say also that F is upward close. Okay. F is upward, and 2, A, B in F, they exist. C, smaller than A and B, such that C is at F. So this notion here is what we could call the prime filter on P, okay? It's a prime filter on P. And why is this true? It just follows from these conditions. You just... a character will be on D of P, will be... If you restrict the character to P, you will have a map from P to 1. It will be the characteristic functions of something satisfying this condition here. And if you look at this condition, you see that it will be the characteristic functions of the prime filter. So F is the set of X in P, such that P of X is 1. Okay. Sorry, what did you do here again? Okay. I'm looking... Now, first, I'm describing the characters of this frame. Okay? The character will be a frame of morphism. Now, the frame of morphism, if I restrict it to P... Okay, so I think the restriction to P... P, you have the generator. The restriction to P will have to satisfy the relations that I have written here because it's the case where A is the... is bracket 1. And this phi is actually a phi prime. And actually, I'm looking at the restriction here. And it must satisfy this condition. And if you translate this condition, you get these all. Okay. So, in general, when you construct a frame, you may construct it from generators in general. So you have a frame A, and you would like to have a presentation describing the frame as a quotient Q. From some... these are very easy to construct. The D of P. This... so you should know... Of course, this frame is not free. Okay? But you need also a relatively topology, possibly, in addition. So that's the... I mean, you need to take a quotient and then to have the morphism here. So that's more or less the general constructions of a frame. You pick some generator, linear generator, because you want to have this as a subjection. So, some elements having the property that every element is a supremum, linear generator. But these linear generators should satisfy some relations, which are the... okay?  And the topology, very often, is a bit less... not as general. I said D of P. And so having a relation here, you may as well have a relation here. Okay? So it's a relation between the lower sections and the elements of P. And this relation, pi, let's say A, is S cover. Okay? So S, normally, S will be... contain a sub... a subset of our street. And you want to say that S cover A. And that's the... or phi cover. You see that? You introduce a covering relation like this. You want to say that certain lower sections are coverings the generator. And then you need to say that some stability condition that if S covers A, implies that S intersection B, B, B for all B smaller than 8. Okay? That's the stability condition. And then the notions of sheath is defined using that. And that the notions of sheath mean something like this. That T in D of P is a sheath. And for all S covers, by covers, S in T implies A in T. Okay? That's the sheath condition. Okay? So that's the picture for the two of friends. And it happens that this picture just repeats it completely for two volts. It's the same thing. Except that you have to be a bit careful. There are morphisms around, etc. Yes? When you start with a botanic topology and one defines the sheafification. Yes. One is obliged to do it in two steps to repeat twice the sheafification. Okay. What is the analog here? That's a good quote. I was hoping that nobody would ask me this question. You see, I'm a little bit cheating here because I'm making things simple, which is correct. I mean, this is correct. From a conceptual viewpoint, you would like to know how you construct this sheafification. Okay? I said nothing here. Because what I did was to say what is a sheaf? And then I proved formally that there's a best approximation given something which is a pre-sheaf. There's a best approximation as a sheaf, which is I call the sheafification. Okay? So I don't need to worry. The definition is there. But of course, you may want to construct. And then you do as usual. You do as usual. In other words, in order to be able to construct the sheaf, you need to put some extra conditions on the topology. I said nothing about compositions of covering here, which was important in the definitions of what goes in the topology. This is some kind of pre-topology. Okay? So if I want to construct the sheaf, I will need to have an extra condition of the coverage to be close to the compositions, and then I just repeat what we do this way. But what I was asking is, I suppose the same difficulty arises already for locals, that you have to repeat twice. No, because in the case of sheaf, no, I don't think so. Because in the case of sheaf, the first time you construct a separated sheaf. But if the sheaf is separated, you get the first pre-sheaf. If the pre-sheaf is separated, then you complete it. But if it happens that the pre-sheaf is separated, then there's nothing to do. But here everything is separated because all the diagonals are triggered. So in general, to construct the analogue of the composition, the work of covering families in the Rottenbeek topology. Yes. So in this framework, you have to make a process of the composition, but you have to iterate it. Yes, exactly. If you have a cover like this, you may think of this as a family of lower bounds of A that covers A. And then if you want to pull the compositions, you just... So the same thing is happening in topology. So first, you have the solutions of the linear problem. This is really free over a small category C. You take the pre-sheaf, and this is the free co-completion. Now, you may want to take a quotient of that. And how do you do a quotient? You pick a family of arrows in C. In fact, you put it in PSC. You pick a family of maps between pre-sheafs or a set, a set of arrows in pre-sheaf. And then you say that something, five, C, X does not see the difference if the map is invertible, is an isomorphism, for all U from A to B in five. So that's the condition that X does not see the difference, not between A and B, but the object X thinks that U is invertible. And then it happens that if you look at the full sub-category of PSCR, the full sub-category of the pre-sheaf category, those objects are probably the five-sheaf. Then this thing has a left-hand one. This is a complete sub-category, and it has a left-hand one, maybe L5. It's sometimes called localization. And it's the solution of universal problem. So you compose with the Unida polka with L5. If you have a function here that inverts all the maps in five, in the sense that, well, you have to say that F over three is a few and invertible. F over three can mean the continuous extensions of the function of the pre-sheaf category. If this is invertible for every U or U in con, then there would be a function here that is co-continuous and invert, and it's the solutions of the problem of inverting all the arrows in five. And this category is what we call presentable, or in the language of Gabriel and Ömer, it's called locally presentable, but today people have started to call that presentable category. So locally presentable or presentable categories are the same. This is all presentable categories, so-called presentable categories are constructed this way. You take a small category C, you take a small set of arrow, and you do this process and you get, so it's a kind of generalized Topol's theory in the sense that you just get here a category with co-limits, with generators, and that's essentially it, there is no other property than that. The generators are small, so-called, and... Excuse me, I didn't understand just... Can you start from, in this localization, you can start with what you call, instead of fresh sheep of sea, you just need, this is what you call a presentable category, and then you take a set of arrows and you can do this, or do you need to start with a pre-sheep? With heat? To do this process of... Oh, yeah, yeah, yeah, I did it for pre-sheeps, but you could start with a presentable category, start with a set of maps here that you want to invert, and localize, and you have the same theorem. Yeah, you don't need to localize from a set of maps in the category of pre-sheeps, you could localize a set of maps in a presentable category, but the presentable category are somehow defined by localizations of pre-sheep categories. You have actually found some compatibility. No, no, it's completely arbitrary, yeah, that's the good side of it. There's always, you start with... That's it, that's a theorem of the Gabriel energy. So that's the Livia aspect behind Topos theory. These are like avian groups, okay? But then you want to know when is this a Topos, okay? And that's where the so-called growth in the topology happened. So our, all the notions of sight. So this growth in the topology here is the following data. It is a set of monomorphism, S to the representable Y, A. So I'm choosing monomorphism this time, because I want that the localizations be a Topos, and this is called covering seed. A is a representable, A is in seed, and so this S is actually a sub-pre-sheep of Y, A. So you start with the collections of monomorphism like this, and you suppose that the collection is stable under pullback, so condition stability. So that means that if you have something in, if you have a map F and A to be in your category and you pull back the cover, the pullback means to be in fire. This monomorphism means to be in fire again. And now once this condition is satisfied, if you invert universally this class, you get a Topos. And this is called a sight. You don't need that the monomorphism are stable under composition. You need that only if you want to construct using this sheafification. In other words, what happens is that if something is a sheaf with respect to a certain class of monomorphism like this, it will be also a sheaf with respect to the classes that you can obtain by composition. To be a sheaf does not use the fact that the class of monomorphism that you choose is closing the composition. It's only about the constructions of the sheaf where it is useful. It's like in ring theory. You can divide a ring by an arbitrary set, meaning dividing it by the identity of the final set. Yes, right. And it has similar universal property. Yes. It's a universal ring which kills all the elements we are given. Exactly. Exactly. Yeah, yes. It's very, very in the same picture. Yeah. Okay. So, you guys said enough about, let's see, how much time? I still have 25 minutes. For the rest, I'm going to talk about the logic and topos theory. Olivia said something about it, geometric logic in her course yesterday. So, we'll see more today. But I was beginning with higher order logic, which is a different subject in some sense. So, this is a topos. Let me say, okay, a topos, okay, a topos. I'm defining, I'm giving a new, not a new, but yesterday I gave the calculations of what a group with topos is in terms of the zero axiom. Right? I described it. Now, I gave another system of axiom. It's a presentable category, such that, which is locally cappes and flows with sub-object classifier. It means, locally presentable, I'm sorry, locally cappesian. E is cappes and flows, founters, if it is cappesian, so if it has products, founite products, and the founter has a right adjoint. And the right adjoint is denoted as an exponential. So, the right adjoint is denoted as an exponential by A. And the right adjoint is an internal arm? Yes, right. So, you have that A plus B, you see there's a B to C to the A. Right? There's a bijection between these two kinds of lines. And locally cappes and flows mean, locally, it means that, in addition, all the slice circles, if you look at all slice categories, they are cappes and flows too. So, it means, for example, that there are pullbacks, because if this has product, you mean to have pullbacks in E, or further product, okay? Now, the sub-object classifier means that there is a universal, a universal, so this universal monomorphism, the target of it is a term, and the source turns out to be the terminal object. This needs to be proved, but this is, and the point is that for every A and every monomorphism, there exists a unique pullback square, a unique path such that this is a pullback. A nice way to express the universality of the monomorphism, U, is to introduce the category, the category of monomorphism and pullback. So, you look at the category where the objects are monomorphism, and the objects are the monomorphisms in E, and where the morphisms are the pullback square. You look at this category, the category of pullback square, pullback squares of monomorphism, right? And this thing is saying that this category has a terminal object. This is the terminal object of the category of pullback square of monomorphism. That's one way of describing the universal property of the monomorphism. Okay, and... Yes. So if I understand this definition well, it's not the topos of photonyx, it's the topos in matter, doesn't it? No, this is the group of the topos. Presentable. And now, on the students that you could do a lot, just by the axioms, that it is locally presentable and a sub-event classifier, I mean, most of the constructions are many, many could be done, so you drop the presentable. You just start with the category, which is locally presentable, so the sub-event classifier. And that's the notions of elementary. And this is a really, very, of course, a very nice, important discovery, because, let's say, Michel Biennial saw also, afterward, that you could interpret set theory, enthusiastic set theory in any elementary topos. In other words, you can use the usual notation of set theory to work in a topos. You can, for example, the power set, the power set, what the power object is, which is defined to be a definition to be omega to the n, okay? You could say that omega to the n is subset of a, that's the power set. We know that the p to the n is an object in a topos, and we know that it's not the set of subset. However, using this Bire-Boumice language, it shows that you can interpret this thing as meaning that, and as well as all the axioms of higher order set theory, and it's a model of set theory. Because in usual set theory, you can form higher and higher cardinals, so it's like bed, outflow, and so it doesn't exist in, I don't know if that's the right way. Yeah, yeah, so there are some small restrictions of this kind of set theory, for example, you may have a natural number, and then you have p of p of n, and then p of p of n, it's all there, except for, it's all there, but you don't have, you may not have something beyond. Okay? Okay. So, but for most, mathematics is mostly involved with this sort of, okay, you don't need to go much beyond that, so it contains a big chunk of mathematics today, right? Yes? Can low ectopause, which is not a growth and ectopause, see now the growth and ectopause in some weaker model of set theory? Okay, I don't know, I would like to know, yeah, okay, okay, so, now, when I, Benevol, first explained that, I think, in the meeting in Montreal 1974, and I was very excited. I had to work with geometric logic before, but this was not geometric logic, it was the whole higher-order logic, because it was showing that you could do ectopause theory without any category theory, that you could somehow, it's, the point is not that you should not use category theory, but you don't need to use categorical notations all the time, okay? You just use, make reasoning that are just like classical, and this is very good, because sometime in category theory you have a proof and you have a new diagram, arrows are moving all around, and it, and you say, well, what is this, it's difficult to understand you have, or at least I find sometime difficult to understand, but if you succeed to translate what this diagram is saying in this language, sometimes it becomes obvious. So in other words, there is an alternative to understanding. It's not that you should not use category theory, but sometimes the classical notations are more comprehensible, and therefore you should use it when it is the case. What is the main theorem of elementary topos theory from my point of view? It's the fact that if you take e elementary topos, of course the topos doesn't need to be elementary, it could be a topos, okay? But what I'm going to say is somehow through the construction, well, with the elementary topos, then you can do topos theory. You think of e as set theory, now you start again, you do sgfo in e. So for example, you first look at categories inside e, cap e. So what is the category inside e? It is an object of arrows, an object of objects, and a source and target map, and then a composition law that will be like a map like this, you see one. Everything that you can do, usually in mathematics, you can do it in a topos, and therefore you can do cap e, okay? And then there's notions of a site, you can explain what is a site, and you can do topos theory inside a topos. In sgfo what they do, I'm saying that in sgfo what they do is a bit more complicated because they allow to form things like the category of shields of precious, viewed as a site, but it will be a large site, so they use a larger universe, so they use more than what you allow, because in some proofs they need to say we reduce the case of precious, which is the use, which is this site, so for example in topos theory you need the category of set, because you want to have precious, but the category of set is actually a big category, and therefore doesn't they use the universes to make it smaller, and you have always this problem that you have smaller categories and big categories, and usually a pre-sheaf is something in a big category, okay? The same thing happened here, in the sense that you define what a pre-sheaf is, it would be a family of an object over c0, okay, because you think of xa is the union of all the values of the pre-sheaf, a in c0, you think, okay? A pre-sheaf is like if you first a family of set, and then this family of set has an action, right? So you take t of your star of x, and then you will have a map, which is the action to s of your star, so it's, you see, like a descent condition and etc. You can explain what is a pre-sheaf, okay? But this pre-sheaf, you will have a category of pre-sheaf, but this category of pre-sheaf will not be small, it will not be a category of like a key. So you cannot do the game of looking at, like in Nigeria Power, they look at, and Glotendic topology is like on the category of pre-sheaf, so like it, like, they can say that the purpose is also obtained from the category of pre-sheaf, the canonical topology, and if the shapes and things of, you cannot do this. I'm sorry, maybe we can discuss it. Okay. Yeah, so anyway, you have pre-sheaf on a small category c, c is a small category, okay? And this is an elementary topos again, okay? And you can add a topology and you will get a topos. And this topos will have a geometric morphism to e. It will be equipped with a geometric morphism to e. And the theorem, I would say kind of fundamental theorem, or at least one fundamental theorem of elementary topos theory, is that you get all the category of toposes over e, the toposes over e, the topos that are equipped with a geometric morphism. Okay, so there is a category of toposes over e that is equivalent to all construction from internal things here. So I would say that the same thing as toposes in e. It's a bit vague here because I have not really explained what a topos in e is, but it's essentially given by a site. A site is certainly a small object in e, and you can describe everything in terms of internal site, a small category in c and with a topology. So this is kind of amazing. I mean, look, we have a topos which you take as a space. I'm drawing the topos. And then you have another topos, let's say e, and over it, okay? By mean of a geometric morphism. By mean of a geometric morphism. Well, this whole thing over it is completely described by a small thing in e. So should I read that equation as elementary toposes over e are equivalent to internal growth in the toposes in e? They are given by internal site in e. So you can prove things about toposes over e by working in e. So a geometric morphism of toposes in the case of elementary toposes, what is it? Is it a pair of a joint factor? Yes. Let's say e is a geometric topos, so-called, I prefer to call it a topos than a geometric topos, right? And elementary topos, I call it an elementary topos, right? I think that's the right thing to do. So e, let's say, is a topos, that is a geometric topos, f is a number of topos, this is a geometric morphism, and you can, okay. Sorry, so could you explain again the result, but I don't get it. Yeah, yeah, yeah, yeah. This is a bit, a bit vague at the moment. I just want to essentially say, I'm just saying that if you have a geometric topos over e, then you can construct a site internally in e, such that the category of sheaves on that side, on that side, would be, as a more quick way, you have a cumulative triangle type. The first sheet is that quantified part from c to e. Yes. And when you do the notion of a, of a, of a quadratic topology, you have to take a set of c's, and do you take this set of c's in the sense of set, even in the solution e or a set of c's? Yeah, I'm, yeah, I'm sorry, maybe we can discuss that after, but I don't have much time left. Two minutes? Three? Okay, okay. Okay, so in particular, you can do the theory of locales in e. I mean, some toposes are very simple there with locales in e. So if your topos in e is a local, you get a special kind of topos over e, which is called a localic morphology. If your topos over e can be generated, can be described by a local inside e, or if you like a frame inside e, there's a theory of frame. The whole theory of frame that I described to you can be repeated in e. Everything I said is complicated through again in e. And it give rise to the notions of localic morphism. Now, there is a topos which is often denoted like this. It's a classifying object. If the topos freely generated by one object, I mean, I denote it like this because it's like a polynomial ring. It's in the category of algebraic topos. In other words, if you have a topos e and some object u in e, then there would be an algebraic morphism by algebraic that takes biocits to u. And this topos has a very simple description. It's the category of covalent topos from finite set to set. It is a pre-sheaf category. It's pre-sheaf on the category opposite to the category of finite set. And it's kind of interesting to look at locales in this thing. Locales, I'm sorry, a frame in set x is a covalent punter, a team from a finite set to frames, so the category of frames. A frame inside this topos can be described as a covalent punter from the category of finite set to frames. And this covalent punter has various properties. So, if you have a map from end to end, then you have a map from Tn, a map of frames from Tn to end. And this is Tf. And Tf has a left-hand point, which is f from Tn to Tn. And as well as other things. But the point is that T is a geometric theory. This is the kind of theory that Olivia was describing yesterday, but it's the purely algebraic view on it in the sense that Tn is a set of formulas of x, phi, that's the notation. With the variable phi, x contains in the first, you have an infinite set of variables, so you look at, suppose that the variables are contained between the first x1 and phi xn, and you look at these formulas, and you take the logical equivalence, relation given by the heraxioms, and you, okay, and this would be a frame by constructions of the logic, of geometric logic, and it will satisfy all the axioms, and it will be a frame inside these totals. So what this thing seems to say, I will finish with that. Same tactic, a category for some geometric theory that we are to discuss, you say that this category set of x is a syntactic category for some, for geometric theory. This is not the classifying totals. This is classifying totals for one sort. If you have two sorts, you take two objects, that's the only difference, right? But the point that I want to make is that topology maybe can be understood as the study of frames. That's one interpretation of what general topology is. Okay, so now we can develop general topology. There are all kinds of notions, compact spaces, all kinds of theorem, but we can do it inside the totals. We just repeat, but it's not always clear how to do that because the axiom of choice, we can't do that. This is topology without the axiom of choice, and then as a consequence of that, we get theorems about theories, okay, geometric theories. So I think this is maybe a version of the duality idea that is expressed by Olivia in duality between theories or maybe quotient theories, okay? This is a very specific. Yeah, or maybe it's a version of duality or maybe it's an extension of duality. I don't know exactly. But it is in the same picture. I don't know how... I don't know, it's just an idea. Thank you.