Philosophy Now Forum

by Matt Qvortrup

https://philosophynow.org/issues/156/Philosophers_on_Baths

Archimedes famously said Eureka! We all know that, and we might also remember that his exclamation came when he was stepping into a bath and realised that the volume of water displaced must be equal to the volume of the part of his body he had submerged.

Here is an example of cause and effect that is not based on a constant conjunction.

How do you "realise that the volume of water displaced must be equal to the volume of the part of his body he had submerged" ?
You couldn't really try and measure it before you realised it should be the case. Before you realised it should be the case why should you think that sort of measurement was useful?
It probably isn't exactly true, because the parts of body submerged will also be compressed. This is apart from cases where a submerged object dissolve, or acts like a sponge, or a purely logical possibility happens such as the part immersed becomes a different object, or disappears. Then the bent stick phenomena might make it seem, from visual experience, that the volume of a submerged object should increase; even if the water level doesn't correspondingly increase.

Hume "There is no object, considered in itself that can afford us a reason for drawing a conclusion beyond it". But it seems to me that we don't want to have to go beyond objects in our understanding of causal situations. We don't want things to appear from nothing, ( "nothing comes from nothing") or disappear into nothing, We want to be able to trace factors around in causal situations, so that all alterations are a rearrangement of the same old continuing stuff. This is because if we don't have to go beyond the same old continuing stuff as the situation develops, just because of that fact. the continuation of the same old stuff must appear sufficient to bring about that development of the situation. And, being naive realists, we naturally want and expect the contents of situations, themselves, to bring about what happens in the situation.

Hume's statement was aimed at any attempt to make a logical deduction from the idea of any state, to the idea of any other, and in turn, I suppose, was aimed at any vaguely Aristotelian notion that what happens in a situation results by a logical deduction from the essence of the substance involved. But this notion would seem to require a logical deduction from one object, substance, or essence, to what happens to another such thing, who's idea would be logically outside the idea of the first. Billiard balls seem to provide a good illustration of this point. But on the contrary, we are not trying to make a logical deductions from the idea of any state, but are making comparisons to see if we apparently need something new, or the continuing same stuff can bring about the result.

Archimedes had also to realise that a heavier substance will have less volume for the same weight as a lighter substance. Consequently he could tell if the crown was made from a heavy substance, gold, or something else if he knew how much it weighed and what its volume was. So he needed a way of measuring its weight and its volume. And the way of measuring is volume was to immerse it in a fluid, and note how much the volume of the fluid increased.

In the context of measuring weight, Archimedes himself is also famous for his demonstration of the law of the lever, which uses a way weights can be compared, but we could also describe it as producing a principle for comparing weights. But all this only works if weights and volumes continue. According to the axiomatic method we are supposed to list all our assumptions at the start of our demonstrations, but Archimedes does not mention this assumption. So it seems he must have overlooked it as something that is too obvious to mention--this is surprising.

But further, in passing, raising the water level, or the level of a fluid, IS raising a weight; the weight of the fluid raised. And the distance a weight is moved is work. We can also raise a larger weight a smaller distance, by arranging for a smaller weight to move a larger distance in the opposite direction, as in the law of the lever, or as in hydraulics. And this seems a principle made use of in pulleys (pulleys don't necessarily move a large distance in the opposite direction to the direction of the load, and also the screw) and the Archimedes screw. (Also, it seems Aristotle held the law of the lever was linked to the perfection of circular motion, but if raising a fluid weight by submerging a weight is a balance, like a lever is a balance, this does not involve circular motion.)

But all this is much too quick for philosophy, and for rigorous mathematical demonstration. Archimedes and his bath involve comparisons and continuation of volumes. It also involves the comparison and continuation of properties. It will be no good if the weights of objects continually change for no reason, and it will be most helpful if they continue unchanged. The volumes wont continue if their objects dissolve, compress, evaporate, or are porous. Because we are only considering volumes that don't change for no reason, or for any of other reasons listed. This can seem as if we are making it definitional what sort of objects and properties we are considering. But this is a confusion, because definitions don't make things happen. On the present view we are restricting ourselves to a particular sort of object by our 'definition', but it is the existence of these sorts of object together in a situation that makes the result come about, not a definition.

As an example of issues involved in deciding what shape something has, and so what volume it has, let us ask why we think a round coin is round? After all a round coin will look lozenge shape from nearly all angles. I think this is because we don't want to draw conclusions beyond objects, as situations unfold, or develop, we want the same unaltered object to continue through the various variations we may notice regarding it. To this end, if we supposed the coin was lozenge shape, then although we could manipulate a lozenge so it looked round, unless we used lenses, the round size would appear smaller than the lozenge, just as a lozenge is smaller than its round coin, because the perspective and angle cut the apparent size down. But on the other hand if the coin is round we can easily account for its various lozenge appearances by perspective and the changed angles from which we look at it. And this sort of thing can be checked using geometry, others testimony, and comparison with other objects that are supposed to remain the same. This sort of principle will also be involved in judging if a lever, for instance, is symmetrical, although the view supposed is normally one where it is obviously symmetrical. (wrong again!).

Although it doesn't really apply to the normal experience of a coin, geometry can show why objects appear to change relative position at different distances from the observer, if they remain the same object. For example, the moon seems to follow you along if it is near the horizon and you move, say in a car. This is because the further away an object is the smaller the angle it changes for any distance moved, as you can see by a taller or shorter triangle with the same base. The taller it is, the smaller the angle at its top. So the moon, being at the top of a very tall triangle, does not seem to move relative to yourself although you move a large distance, and although the roadside scenery, being a lot nearer speeds past, showing large changes in apparent relative position.

Hume would try to explain this by coherence, and the way the mind runs along the different possible perceptions of such a coin, these principles of convenience making us regard the differently perceived states as one object for pragmatic reasons. But, on the contrary, if we don't have to go beyond the same state as that varies in its appearance to us, this claim does not have anything to do with the imaginations principles of connection.

I can further illustrate how we don't like things coming in and out of existence, because we don't understand it, with the childhood trick two little dickie birds sitting on a wall, where bits of paper stuck on fingers are made to disappear and re-appear, by swapping protruding fingers. Also the magicians trick of pulling a rabbit from an empty hat seems to work on the principle that obviously an empty hat is insufficient to produce a rabbit, and the case of the miracle of the feeding of the five thousand is similar. But it is not just that we don't like this sort of thing. In these cases we can't just suppose what is already there produces the difference, so the developing situation, apparently, can't be self sufficient. But if on the contrary we can see what happens as the result of factors already present, continuing, such situations will appear self sufficient. If situations do appear self sufficient then they can't appear to need to depend on anything else, so they will appear to be independent of ourselves and 'our understanding'. Nevertheless mighty oaks grow from small acorns and so the natural world can seem full of cases where the situation is miraculous from this perspective. It seems possible that some attempts at understanding are attempts to meet this apparent challenge of the discrepancy between situations apparently not appearing themselves sufficient, and our incorrigible naive realists objective to see how they are. In this way we can have an objective that is not based on any guarantee, or probability as to what will happen, although success towards this objective will give a reason as to what further should happen and why.

Another example illustrating this aim of seeing how a situation is self sufficient is provided by evolution through natural selection, which, especially originally, wasn't very good at providing any predictions. But because it showed how the natural world was itself sufficient to produce life forms and their relationships as currently observable it made the need for any grand designer god to account for these things, redundant. And explained the phenomena just by this 'showing how the environment was itself sufficient'.

Lets start again by noticing how to compare weights, and how we can know if any object has a weight. It seems that a weight can only be known by its effects . Such effects would be by squashing or stretching other objects, or by balancing other objects (or weights), or by feelings of heaviness when we try to push, pull, or lift an object. This last set of cases seem different from the others in that they involve qualitative feeling, perhaps with some movement, while the others involve quantities of movement only. Feelings and motions seem, on the face of it, completely different things; we could for instance see something move without any feeling being involved, and we can probably imagine, or dream, a feeling of weight, or heaviness, without seeing anything move (or close our eyes). Nevertheless, it may be that feelings of weight are a natural sign for the sort of interactions an object will have, just as words can be artificial signs for things completely different from the words. If the feelings of weight were such a naturally occurring sign or indication for how an object will behave, this might be very useful, and also very convincing as to how the object will behave. But for all that the connection would seem intellectually unsatisfactory because the feeling will just be found conjoined with something completely different--a motion--and how could we know that on other occasions it will also be found like that? Or why should it be the case if things are found like that?

These effects that I have mentioned are sorts of happening to other objects that are found to occur in the presence or application of an object. This is the sort of conjunction that Hume requires for his 'constant conjunction' theory of causation, except that we don't need the 'constant' element in that theory to supply the motivation or cause for our supposing the conjunction will happen again on a new occasion. And we don't need the constant element to supply the motivation for a universal idea, or law, from which we can deduce what will happen in a new instance. --I don't know whether this may seem a disastrous move from an empirical point of view, which is why no-one makes it.-- But this is because we are TRYING to see how that very object that was involved in such a conjunction, could produce, by continuing to act in that way, the occurrence in this new situation. This is so we can make progress towards our (incorrigible naive) objective of seeing how the contents of the situation could themselves be sufficient to bring about what happens. It is true though that we often can suppose how objects act will be as similar objects have acted. In this sort of case the objects will not have continued into the new situation being considered. But a different reason we might suppose a similar object should be found conjoined with a similar effect is, again, derived from thinking that the contents of a situation should be themselves sufficient to produce what happens. Because, if we do tend to think in this way we may also feel that (even if we don't know how they do it) if the contents of a situation THEMSELVES produce what occurs, and if we suppose we we have another situation with identical contents, then it should produce the same results, or there should be some difference between the two situations after all--which will give us a motivation to re-investigate both situations. Or else we'd have to admit we can't see how both identical situations could themselves produce different results (of course, it would not be logically impossible for identical situations to be found giving different results.) A related basis for inductions of this sort are arguments from symmetry. For instance, in the law of the lever, it is supposed that if both sides of a lever are symmetrically identical to each other, they should balance, as there could be no reason why one side would fall or rise more than the other. Philosophers don't normally seem to mention this sort of thing as a way of reasoning inductively, which apparently gives a reason why what happens on one occasion should happen on another and so does not just assume that the past will resemble the future. To repeat the reason is derived from causal realism; that the contents of a situation should themselves be sufficient to bring about what happens, and we can't see how this could be the case if identical states issue in different results (as with symmetry), even though that is perfectly imaginable from the point of view of what is logically possible.

There may be several reasons why philosophers never mention such a basis for induction, although it at least seems different from supposing the future will resemble the past. Firstly, perhaps they think if symmetrical arrangements produce identical effects this is a prime case of something that must be learnt from experience. I agree that experience can show similar states don't produce similar effects. We could also find it easily the case that similar states--or symmetrical states-- are obviously producing similar effects. But none of this shows we can't suppose symmetrical arrangements should produce similar effects because they should each be self sufficient, but how can we produce a different result when there is nothing different to produce it? Secondly, they don't think we can make sense of anything itself bringing about anything-- but the basis works even if we don't know how anything could actually itself produce anything. It relies on having any chance to to progress towards the objective of seeing how the contents of a situation could themselves be sufficient to produce what occurs. Thirdly they are looking for some increase in probability as to what will happen, or at least some probability for the alternative basis for induction. And fourthly it must seem a bit a-priori, and a-priori principles that have any meaningful content seem un believable and to raise questions about themselves that are just as awkward as what they are supposed to resolve. But symmetry, or causal realism, doesn't come with any guarantee, or probability as to being true. It does not come with a knowledge that it is true. But, if it were true that would give a self sufficient reason why e.g. symmetry, should be true, and should be a useful basis for making inductions. (This sentence may seem circular but if there were a self sufficient basis for something to happen, that could explain why it happens. That is not circular. Symmetry might be one of the things explained in this way. That is not circular. Symmetry might be used to judge a degree of self sufficiency, as what is symmetrical should in one case produce what is produced in the other. This is about the degree of self sufficiency apparent. Perhaps exact similarity, or perfect symmetry in constitutive reductions, might be as far as we can get towards apparent self sufficiency.)

But it may be that philosophers don't recognise symmetry as a different basis for inductions. Perhaps they think 'One side acts in a particular way, then we suppose we have another side that looks symmetrical to the first, so on the basis that a new instance will resemble a previous instance in a respect, when it resembles it in another (i.e. that the future resembles the past), then the second side should act the same as the first. So 'symmetry' is just another case of supposing the future will resemble the past.' However if we suppose there is a self sufficient reason for something to happen then it is not just the similarity between events that makes them act, or result in the same thing. A self sufficient reason for something to happen is different from a similarity of occurrences, although, if present, it should produce similar occurrences on similar occasions.

Hölder, Mach, and the Law of the Lever: A Case of Well-founded Non-controversy (openedition.org)

The above links to a discussion worth considering involving the nature of proof in mechanics, focusing on the proof of the law of the lever of Archimedes. But the discussion centres round the adequacy of the proof (or reasoning)' whether or not it assumes what is proved, or the assumptions contain what is proved; and the origins of the premises/assumptions, i.e. are they known a-priori, or derived from experience. However my question is "does the reasoning show how objects continuing to act as supposed WOULD act in the way "deduced", or described?" --I'm not concerned whether our 'knowledge' or thought that the objects have weight and continue 'with' it--continue to exhibit it, is known a-priori or by experience, but whether, if they do continue to act like that together in a situation, what the result would be. It is difficult to separate out this question. They are looking at origins and logical certainty, I am looking towards the result of the origins continuing to act as originally, in various situations, in which case those new situations can be seen as self sufficient in terms of that continued original behaviour.

Mach does not seem worried about allowing that things being symmetrical should produce the same results (balance). What he finds raises the question of 'how we could know that?' is when the proof moves from the symmetrical arrangements of weights along a suspended or balanced beam, to an A-symmetrical arrangement at reciprocally differing distances from the original balancing point. But, if weights symmetrically arranged around a point balance at that point then we can take any sub group of the originally arranged weights, find the point around which they are symmetrical and they will balance at that point. Since they balance at that point, that is the point at which their combined weights in effect act, just as the distributed weights on the original symmetrical beam, in effect, act as if combined at the centre point of the beam. (this 'in effect' just means, if we weigh this contraption supporting it at this point around which it is symmetrical, the result will be the combination of all the distributed weights, as if they were there.) If we now combine this subgroup at their sub groups balancing point then this shouldn't make any difference to the balance of the original beam because that point is where the uncombined group was (or is) in effect acting anyway. Although our randomly combined subgroup may appear to make an A-symmetrical distribution on the original beam.

But now I have tried to put this in a logical argument I suppose it will be thought what happens in the case depends on logic. But it doesn't, it depends on things continuing to act in a way. From their existing at any time, or acting at any time, nothing logically follows about their existence or action at any other time. And it isn't our having a universal idea of their existence or action that we can apply to a new time or opportunity for acting, that makes what happens occur, but their continuing in the new situation, with those properties.

objectives;

1) To understand how the contents of a situation could bring about what occurs

2) To produce successful predictions of what will happen

3) To find axioms that are certain from which everything else must follow

All these three seem distinct although pursuing 2) might incidentally result in something that seems like 1) and visa versa pursuing 1) might produce an effective way for making predictions. On the other hand 3) is liable to seem a priory, and proof that 3) is impossible has been taken to show that 1) is impossible. (And, independently, the causal theory of perception may be taken to show 1) is impossible.) 3) seems the most systematic way to present the understanding of a subject. However that is; pursuing 1) is working towards a goal, as is 2), it is not necessarily, or obviously, working from axioms.

Pursuing 1); Going by Hume's habit hypothesis, we must be able to see if several occasions appear similar or not, or be able to tell what would happen on a new occasion if it is to be similar to what we have experienced. --It would be no good trying to suppose that the future will, or does, resemble the past if we can't tell if the future resembles the past or not. So, if an object behaves in a recognizable way on one occasion, we should be able to tell what would happen if that object were in another, similar situation, and behaved in that way. This may not give anything like an ultimate basis, or real being, for why what happens does so, but nevertheless we may see how the conglomeration of properties that constitutes an object on one occasion could produce what happens on another occasion, if it were in that situation and acted similarly, and without this being produced by habit, but from the motive of trying to see how the sort of thing found in a situation could produce what happens in a situation. This seems distinct from, but not inconsistent with a supposition that an object continues into a new situation, or from supposing that a indistinguishable situation should produce an indistinguishable result, or we wont have anything to work with to suppose the contents of the situation produce what happens.

For example, when geologists see particular structures on the surface of Mars they hypothesize that that is what would be produced if water had flowed on the martian surface. Or if astronomers can measure a dip in the brightness of a star they can suggest that that is what would happen if a planet orbiting the star passes between us and the start. This differs from using a habit to decide what will happen, or what we will believe, It is instead making comparisons and trying to match some actual occurrence with another actual occurrence or supposed situation.

It seems to me that according to the linked item Holder over complicated things. The linked commentary, in the second post here, says his position is not clear, but suggestive. I suppose, in contrast I may be over simplifying. But anyway:
This idea of the premises including whatever is agreed to be validly proved; Let's take as an example Lego bricks, and what can be constructed from them. If you have ten Lego bricks do you have to have included in this idea all possible constructions you can make using them? Surely not. But all the same, anything you can construct using them doesn't use anything else (I immediately think of 'space' as an objection, but am taking that as given.). So anything we can construct using them doesn't go beyond them. If we say 'it must be, or is, contained in their idea' this, puzzlingly suggests it is present in their collective idea, but then how? And where is it, since we don't have to think of any combination when we see ten Lego bricks? But if we say that any constructions are synthetic, and not analytic on the idea of ten Lego bricks, this seems to imply that there can't be a demonstration of how they can, or would, construct one of the options they can construct. And if being analytic is required for any demonstration to be adequate it follows that if the demonstration isn't analytic, it must be inadequate. But we can demonstrate any of the possible combinations by showing clearly how the bricks fit together for that combination, that there isn't anything else involved, and that if we fitted them together like that, that is what would, necessarily, result. Although this necessity does not prove the Lego bricks won't disappear or transform at every instant. So, in that sense, the result is not proved.

Similarly, if Archimedes takes a weight on a bar, splits it up, distributes it synthetically along the bar, and recombines it synthetically around arbitrary points, he can demonstrate what should, and must happen, if the weight and bar continue and nothing else is involved/introduced. There does not have to be anything wrong with this demonstration, although it is not found in and is not analytic on our original idea of the weight and bar. It may be described as synthetic, but this seems to imply there is something wrong with it, and that it must be subjective. Similarly, natural selection produces all sorts of species never thought of before they are found, and the ancient atom ism supposed everything, no matter how surprising can be constructed from combinations of atoms. According to the Lego brick analogy, the idea of these items isn't presented with or in the idea of the original objects, but nevertheless it is just a combination of them, and so does not go beyond them.
Kant seems to think that if things are synthetic but necessary this necessity has to be accounted for by introducing another level of explanation, that can't be analytic. But the point of the demonstration is to show the adequacy of the bricks supposed to bring about the result, and the results necessity.

Kant may claim instead his extra explanation is required if something is UNIVERSAL synthetic and necessary......

Meanwhile it seems that it's the analytic that's in trouble (not the possible adequacy of the 'synthetic'), because how do you know the meaning of a word? It would be all very well if meanings were Platonic ideas grasp able before the mind, but there are no such ideas, and if there were, meanings would be something different. But in order to sort out the meanings of words, appeals to intentions are obscure and extensions are arbitrary unless you know, and are guided by the intention. On the other hand appeals to agreed usage are about something known empirically, not something logically necessary. Perhaps the illusion that we know the meaning is kept afloat by our 'synthetic' naive realists understanding of the world, that doesn’t depend on meanings or the analytic? So that Kant's opinion that the analytic is ok but the synthetic is problematic, has things back to front.

There are several possible troubles with Archimedes lever argument. Firstly, if any sub combination of weights symmetrical around a point effectively acts at that point, does that mean there are an infinite number of points where their weight effectively acts? Or perhaps they only effectively act at the centres of possible symmetrical combinations of any actual distribution? But how can the same weights 'in effect' act at several different points at the same time, and without being more than the weight they are? Anyway, this seems a bit confusing (to me).
Another thing is that the distance moved by a balanced weight as its opposite moves should be in opposite proportion. And will occur around different sized circles. But this seems something extra that needs to be proved(why do I think it then?). And does the angle as one side is raised and the other lowered, which changes the horizontal distance to the fulcrum not make any difference? Archimedes proof seems to take the simplest case only, but then it seems plausible and can be checked by experience, so is it really proved after all, or is it a bit proof and a bit experience?--Sneakily, perhaps a bit of a cheat after all?

The reason why distances travelled by balanced weights must be reversed in proportion to the proportions of the weights involved is because we'd either loose or gain something without cause if they aren't. (But, balanced weights, don't really move at all? Don't they rock backwards and forwards?) And the reason for that is that the situation wouldn't be self sufficient in that case. This would seem to be a universal necessary principle, ala Kant, but the reasoning loses its force if it is in turn required, or made, to depend on any condition of our understanding. THAT is not the reason, its own evidence is the reason.

Berkeley seems to have thought that people have imagined they can think of objects existing unperceived, by supposing they can abstract the idea of existence from perceived existences and then supposing they can apply this abstract idea to existences that thus exist, but unperceived. "For can there be a nicer strain of abstraction than to distinguish the existence of sensible objects from their being perceived, so as to conceive them existing unperceived?"
Berkeley does not think it is possible to have abstract ideas, in general, for example, the abstract idea of a triangle, that is a triangle, but no particular triangle. And similarly, he does not think we have an abstract idea of existence, or an abstract idea of an objects existence which we can use to make sense of objects existing independently of their experienced existence.

However I don't think that the idea of objects independent existence comes about in this way. Instead it comes about by trying to avoid drawing conclusions beyond objects in attempts to see how they can bring about what is experienced.
For instance, suppose a screen approached by a ball from side A, that seems to disappear behind the screen, and reappear on the other side B. If the ball continues to exist behind the screen, travelling from A to B then we don't have to draw a conclusion beyond its original existence, and its continued existence can account for what is experienced as A and B. And the balls passing behind the screen explains it disappearance between A and B. This is different from supposing that the balls existence behind the screen consists of a series of occasions where it could have been experienced, if we'd only looked. And it differs from supposing that the mind links or attempts to link these sensations, or possibilities of sensations in a smooth continuum so that passing easily along these ideas it explains what happens. It also differes from supposing we are conceptually connecting this group of distinct states to create one continuing object. It differs because it is not claimed that the mind explains what happens, or that our concept, connecting logically distinguishable, and so logically distinct, states enables us to explain what happens, but that the continued existence of the ball explains what happens. If the continued existence of the ball explains the situation this does not have anything to do with the mind, but is independent of it.

It is not claimed that the conceptual combining of distinguishable states forms a state, or continuing object, which is capable of explaining what happens; even though we can look on every distinguishable moment of our object as a logically distinguishable state, and thereby as logically distinct from all other moments, and any possible way of combining these logically distinct states as just one way of describing the subject amongst all other possible ways.
The continued existence of the ball can explain what happens. The mind (concepts) can explain what happens. These are two completely different claims. The one is about principles that may or may not be acceptable to the mind, the other is about how the ball could independently, itself, do it. Whether it could or not.
But, how is it possible to have a principle for understanding things that is independent of any mind doing the understanding?
By trying to avoid drawing conclusions beyond objects. (But consider what happens if our understanding things works by trying to see how they themselves do it? Since if our understanding depends on principles of our understanding we can't see how things themselves do it.) If we don't have to draw beyond such objects in our understanding, just by that fact, they themselves must appear sufficient to produce, or bring about, what is experienced to happen.
This can be illustrated writ larger, by uniformitarianism, which supposes that the natural world should be explained by the continuation of objects, properties, and processes that are currently observable. This was in contrast to catastrophism, which supposed that novel things should be accounted for by individual acts of creation. Uniformitarianism was one of the general ideas behind evolution through natural selection. Which consequently can suppose a world existing for many millions of years independently of humans, and that does not depend on humans, or their understanding, but eventually produces them.

(This isn't to say there aren't other difficulties with independent existence. )

What it seems is needed for this mode of reasoning is a starting position, an ending position, a rout to get from the starting to the ending position, and ideally, comparisons between the parts of the starting and ending positions which register no difference. (Uniformitarianism adds the added complication of the continuation of processes, but they are not processes that create new matter.)
To take my previous example again; the same Lego bricks can be re-arranged to construct all sorts of different objects. These objects may be recognised as different things, as a bridge, a house, a cultural icon, a thing of personal importance. Each having completely different significance, and some perhaps being surprising and never thought of before. Nevertheless we can see, by examining the parts that each of these culturally different objects are constructed from the same bricks. And on this sense, we are not interested in the significance or descriptions of the constructed objects, or of the Lego bricks, but whether they have undergone any change, considered on themselves, from the starting position to their final position constructing the end object. We are not interested in describing them, but in seeing if we do or do not have to go beyond them as the situation develops. (This obviously seems similar to the ancient atomism.)
In this way it seems mundane that new things can be brought about by the continuation of old bricks in a new order. And this is supposed by me to illustrate how the continued existence of weights and their symmetrical behaviour around a points can be put together or traced to produce results which weren't straight away thought of in noticing the original weights behaviour or existence.

Lego bricks may seem not very dynamic. But they are dynamic since they are hard and the nobs on them fit very exactly into the space at the back of them, so in this way it is easily understandable, if they continue with these properties, how they can be fitted together to form different shapes.
We can examine their properties, how they fit together, at the start and end positions, and see if these properties seem to remain the same or are different at the end position. If there does not seem any difference, those original objects, placed together in the end position would produce the end state of affairs. But we may be able to trace these objects part of the way from the start to the end position, and perhaps there is a rout the continuing objects could travel at the times they weren't observed. In this sort of way we can see how the contents of the situation could themselves, by continuing, produce that end result.

In all this there are the questions: can we reason like this? do we reason like this? Is there a justification for reasoning like this? is the justification adequate, in adequate, or to what extent is it adequate or inadequate? How does it fit in with a wider view of our reasoning abilities?
To take the last; we want to compare the Lego bricks in the different situations, to see if they appear or disappear, or remain and continue. In order to do this we could name each brick and list them on a piece of paper. Then, in the various situations we could tick them off against our list of names and see if there are any left over on the list, or more than on the list, or remain all as listed. But this is a long way of doing it, especially if we are continually trying to make these sorts of comparison. A quicker way to make the comparisons is to take each brick in turn and make a mark on a piece of paper, or bone, or rock, and then, at the new situation, go through all bricks in turn and for each brick mark a previous mark and see if, when we have gone through all the bricks we have also marked exactly all and only our previous marks. This procedure itself presumes that the marks (or names) won't themselves appear or disappear during the process. It may seem difficult to check this, unless the change is really obvious. A way of improving the last procedure would be to group the marks together to make relatively easily recognisable patterns. So we might have; IIIII IIIII II, and check these marks against our bricks. Perhaps we will discover that we have IIIII IIIII bricks, or IIIII IIIII III bricks, in which case we can fairly easily see that the bricks have changed, and there doesn't seem to be a mere continuation and rearrangement of the same old bricks involved in the situation. This might always happen, or to give a practicle example, II bacteria placed on a plate with II bacteria might generally, or always result in IIIIII bacteria after 30 minutes, nevertheless our way of checking would still be working. But on the other hand if we are using this method to compare various groups of our bricks, then IIIII and IIIII II must be the same as IIIII IIIII II if this is going to be an effective way of making these comparisons. It must be more obvious if there is any change in our marks than a change in the bricks, otherwise this procedure wouldn't work. (If we mark a bone and put it in a drawer for a hundred years, then sudden unaccounted differences in the marks on the bone are not going to be obvious, but in this case we presume that the marks on the bone can't just change, or we can tell man made marks from accidental alterations.)

Tally marks are still quite a clumsy way to compare objects, and keep a record of them in this way. And they are clumsy to combine. For these purposes it is easier if we introduce an ordered series of words instead, or body parts. But we can still use this ordered series of words to make our checks and keep an inventory of the bricks, as we were using our tally marks. But if we are using this ordered series of words, which we call numbers, for the same purpose as the tally marks then just as IIIII placed together with IIIII II bricks must make IIIII IIIII II bricks, so 5 plus 7 must equal 12 for the same reason. Not because if you put five objects with seven objects you must have twelve objects. Nor because 5 names all classes of classes of ⁷objects which when combined with the class of all classes of objects named by 7 yields the class of all classes of objects named by 12, (or something), but because we are using these numbers to keep a check, inventory, and compare changes or no changes in a situation, and this method won't work unless 7 plus 5 equals 12, --Except we have forgotten how we arrived at this procedure, and are easily confused as to exactly what we are up to, and why. And numbers may be used for an indefinite variety of purposes. Which confuses the issue even more. Telephone numbers, for example. Or yes and no, in computing. Or to inventory not objects but chances. Or in some cases they don't make any sense. How much are two eggs plus one o'clock? Or in some cases we can force them to make sense but only by inventing a shadowy new property of 'being a number' which we force the situation to fit.

How much are two grains of sand plus two solar systems?--when solar systems contain an almost infinite amount of sand. How many times do two grains of sand go into two solar systems?--But you are not comparing like with like. So, two plus two doesn't always make four then, but only if you compare like with like? How is it then that everyone says that two and two equals four and that this is certain? How much is two bunches of carrots plus two bunches of carrots, when carrots can be giants or tiny? It may be more appropriate to compare them by weight, than by the number of carrots, or bunches because then you are comparing weights, and somethings that are like with like. But house number 2 plus house number 2 may refer to very similar houses, and perhaps owing to an administrative error they are both on Berkeley Road, which however are two different roads in the same town, but nevertheless this does not mean they make four, although like with like and very similar, because their numbers are not being used to see if things stay the same or differ, but, as addresses, to name and point at a particular object in distinction to others.

You are a number two, and you are a number two, does not mean they collectively make four, because although it expressly uses numbers it is actually an insult. Or I might be mock christening two people with the number two, using it as a name.
But although two objects and two objects of the appropriate sort may make four objects, this does not mean two actual objects plus two others will always result in you having four objects: that objects can't have disappeared or appeared for no reason, or may have bred or died. So two plus two equals four may be neither analytically true, nor synthetically necessary. If it is helping to check if objects remain or differ and so if we need to draw conclusions beyond objects, or the contents of situations appear themselves sufficient to produce what happens, and so be part of objective reality, in this sense two plus two can be part of distinguishing objective independent reality. And two plus two equals four can be pointing at a necessary aspect of independent objective reality.
But there is keeping a record(keeping count), comparing, and also prediction. Numbers can be used for various things.

Berkeley's claim that the supposed independent existence of objects depends on the doctrine of abstraction seems inconsistent with the observation that the opinion of their independent existence is "strangely prevalent amongst men". Most men, (and women)who nearly all have this opinion, will be completely ignorant of the doctrine of abstraction (and this is something Berkeley himself claims) so they can't be basing their opinion on it.

But how, in this case, is it that ordinary men and women could understand how objects can be independent, when they are ignorant of my opinion of how they can be, and philosophers either think such a thing impossible, or have struggled to see how it could be possible?

Firstly, if it is the case, as maintained by Hume, that the mind runs more easily along its perceptions when they are resembling, or is driven by coherence and simplicity, then this naturally introduces the opinion that these objects and properties continue and that their continued existence explains our experiences, which then becomes judge-able independently of those supposed mental propensities, which were, and are, never thought by the vulgar, to be involved anyway. But this then gives a standard for objective explanation that can be developed and worked on. Which is independent of those supposed minds propensities. Hume himself admits that if objects continue unperceived they are independent of the perception and that "The continued existence of sensible objects or perceptions implies no contradiction." So we may easily suppose what we perceive continues unperceived. But he thinks that this supposition is false due to the causal theory of perception, or "the slightest philosophy" which maintains that our perceptions have a fleeting, dependent existence, and change every instant. Consequently he seems to think we are forced into being thoroughly inconsistent and into thinking our objects are both independent and continuing and dependent and fleeting. However I just don't think this position is at all incoherent, no-matter how many verbal inconsistencies it may generate. (His view is reinforced by supposing we must be seeking some guarantee or probability in our factual understanding. Guarantee or probability are ways of justifying the understanding showing it to be non-arbitrary to the extent it is guaranteed, or at least is probable. But in stead we can attempt to justify our reasoning by showing how the contents of a situation themselves produce what occurs.)

If we suppose normally, apparently, continuing objects being filmed, then the film, which consists in many fleeting images, which are changed every instant, nevertheless, gives an adequate idea of the continued existences being filmed by means of these fleeting pictures, as it is their purpose to do. From an engineering point of view this is not incoherent at all.
Differently, Kantians seem to suppose that "The world is my (or our) representation";That no matter what may be achieved by a camcorder, or film, we (must) represent the world as independent. But, although there may be "representing" going on, this is not what makes the world known as independent. Instead it is the extent to which the contents of situations themselves apparently bring about what occurs or is experienced, which is achieved by avoiding having to draw conclusions beyond things we perceive, as they continue. So we inhabit a world of continuing objects, who's continuation brings about what we experience. And we understand it by seeing how THEY can themselves bring it about. And so, apparently, not by them satisfying a requirement of our understanding. Consequently, the way they do it may be infinitely complicated, and we get the idea that it may be miss perceived by everyone, because it doesn't depend on, or need to be produced, in any way, by our understanding. Thus we get the view "Strangely prevalent amongst men" not by abstraction, or by applying concepts to blind sensations, or by adding something to blind sensations, but by trying to avoid going beyond them, together, as situations develop.

But, secondly, apart from Hume's alleged propensities of the mind, it will be commonly found that children are puzzled by the apparent bald appearance and disappearance of bits of paper in the two little dicky birds trick; That people do not think anyone can really pull a rabbit out of an empty hat, and that feeding five thousand people with five small loaves and two small fish would have to be a miracle. So they do tend to find having to go beyond what is presented in a situation to get to what results in the situation puzzling. And by extension, where they don't have to do this they wont be puzzled. So effectively they do seem naturally to have the attitude I'm describing. (But, naturally, as soon as there is a motivation for denying something, or maintaining the opposite, people will do so.)