Populations: Arithmetical & Geometrical/Exponential Increase

You can partly get to grips with exponential increase by thinking about what is called The Malthusian Catastrophe.


Thomas Malthus claimed that the world's population is growing geometrically; though food production is only growing arithmetically.

What did he mean? In terms of food production, he was talking about the rate of new acres open to agriculture each year. He believed that the rate was fixed. Thus the food supply works, for example, like this: 100, 102, 1004, 1006.... 1012.... etc. That is, the acreage grew by two acres per year in Malthus's' day. It was a fixed rate of increase which never changed.


On the other hand, populations don't work that way. Basically, the more adults who can have children, the more babies there will be. However, that rate isn't fixed as in 2, 4, 6, 8... etc. This is shown, or at least it was, in terms of the entire population of the world. Instead of 2, 4, 6, 8... etc., it was more like : 1, 2, 4, 8, 16, 32... Instead of the simple addition of 2, we have a number being doubled every time. Thus if you compare 2, 4, 6, 8, 10 with 2, 4, 8, 16, 32, we have a difference of 22 even though both progressions included only four changes. The geometrical increase ends in 10, whereas the geometrical increase in 32. Both rates included only four changes or progressions. Clearly, as the progressions increase, the gap between the arithmetical ratio and the geometrical ration will keep on widening. Or, in Malthus’s case, the population increase will keep on outstripping the increase in food production – resulting in starvation, etc.

Another word for geometrical increase is exponential increase. Exponential increase is applicable to most or all living organisms. It's also applicable to human populations. Another way of putting this is to say that exponential increase is proportion to the given number being increased. With arithmetical increase, it is just the addition of 2 (or 3, 7, etc.) each time. That isn't the case with exponential increase. The larger the number, the larger the increase.


Many arguments against Malthus’s argument have been advanced. That is, even if populations increase, it isn't a necessary, or mathematical, fact that more people are likely to starve. However, even if Malthus’s prophesies were false, which they were (in the UK at least), the exponential increase of families is still (largely) true. Or at least it's true given other (many other) conditions, such as: that all off-spring themselves have families and that what happened to the first family, will happen to all further families generated by that first family. In other words, all the women need to be fertile, all the children need to survive into adulthood, and all the children need to get married, etc. If all these factors occur, then there will indeed be an exponential increase in the population number.


We can say that if all the conditions remained the same for Malthus's arithmetically increased food production, he would have been right.... But that's just it – he didn't foresee the possible other conditions in either the geometrical or the arithmetical case.

However, you can have exponential increase that isn't precise or which fluctuates. (Does that automatically stop it from being exponential increase?) That is, in theory, a couple could have six kids. Those six kids may get married and each have six kids. That would amount to 36 people in two generations. And if those 36 people did the same thing, that is, each have six kids, then the number of people would now be 36 times 6, which is 216 people. So from one couple, and in three generations, 216 people have been produced! That's an increase from 2 to 216 in twenty or so years. Now that's just one family that has produced 216 people in just twenty or thirty years. What about ten or a hundred families?


Take 100 families which have each produced 216 persons in twenty years. That's 216 times 100, which is 21600 people in twenty years. That is, 100 families have produced 21600 people in twenty years.

Hypotheses & Observational Consequences

How are Hypotheses Tested?

According to Wesley Salmon, test their hypotheses by deducing observational consequences from them. This suggests a degree of independence of the hypothesis from observations or from testing/experiments, etc. Indeed it suggests, quite simply, that the hypothesis comes before all observations or tests. An isn't that how the layperson, after all, takes the word hypothesis? He would say that the whole point of a hypothesis is that it's a stab in the dark.


Yet if a hypothesis, scientific or otherwise, were really a complete stab in the dark, what would be the point of it? It would be arbitrary as well as a complete fabrication. It would be plucked from the air and would therefore, surely, have no relevance to anything scientific. For example, I can formulate the hypothesis that the sun is made of cheese. What's the point of it? It's clearly false and clearly scientifically illiterate. Nonetheless, the suggestion seems to be that this is what a hypotheses is – a complete stab in the dark.

Thus we can conclude one thing: even though an hypothesis comes before its observational consequences, it clearly doesn't come before all previous observations. It clearly doesn't come before a whole lot more as well: previous theories, laws, tests, experiences, etc. To use the language of old-style epistemology: an hypotheses is not aprioristic. If it were truly a priori, in the traditional rationalist sense of that term, then it wouldn't be a scientific hypothesis (though I suppose it could be a non-scientific hypothesis).


Anyway, the hypothesis is tested in the sense that the observations it predicts, or the observational consequences of its content, do come to pass.


Now we meet two more scientific terms: confirmation and disconfirmation.


If the predicted observations or events occur, then hypothesis is confirmed. If they don't, it is disconfirmed.


I mentioned that no hypothesis can be truly free-standing. Another reason for this is that the formulation of the hypothesis must assume various or many things. These assumed things have been called auxiliary hypotheses because they underpin the given hypothesis. Or, alternatively, the auxiliary hypotheses are contained within the (new) hypothesis.


Wesley Salmon gives the example of a medical experimenter who predicts that a certain bacillus will be found in the blood of a certain organism. Now in order to be scientifically sure or certain of his experiment or hypothesis, he must accept certain auxiliary hypotheses about the optics which are part of the experimental set-up and which includes the microscope itself. However, in actual fact, the scientist in practice doesn't really need to assume or even accept any hypotheses about optics at this - or any - moment in time. He can ignore them. He's not a scientist of optics nor an expert on microscopes. Other scientists are. What this really means is that both logically and scientifically these auxiliary hypotheses underpin his experiment even if the medical experimenter need not - and probably will not – know a thing about these extra hypotheses: he hasn't got the time.

This is an example of Quine's scientific holism and the scientist concerned need not be aware of the “web of science” or even large parts of it. The web exists regardless of the particular knowledge of the individual scientist. It existed before him, during the experiment and it will exist after the experiment.







Deductive Consequences?

It may seem strange to argue that “a true observational consequence follows deductively from a given hypothesis”. Or, more precisely, it is the use of the word “deductively” that may seem strange. Surely deduction is a purely logical matter? How can anything “observational” deductively follow from, well, anything logical or even from anything non-logical? Surely only theorems, conclusions, etc. can deductively follow from something or even from a given hypothesis. That is an understandable position.


Nonetheless, if the hypothesis has a certain given content, and that content says that if H then O, then if H then O it doesn't matter if the content of O includes predictions about observations, experience or that which is empirical. After all, this is in fact a conditional statement. That is, if this hypothesis is correct, then O will occur. It doesn't matter if the hypothesis, or O, has empirical content, or says that something observational will occur. It says that if it were correct, then there would be certain observational consequences. The hypothesis, or conditional, generates, as it were, what deductively follows from it, even if what deductively follows from it are indeed observational consequences. In other words, the hypothesis is not claiming that there are logically deductive sequences in nature, as it were. It's saying that given hypothesis H, then O (the observations consequences) will follow. The deductive relationship is between H and O, not between one aspect of the world and another (in a non-Humean manner, as it were). Alternatively, there is no necessity in the world, but there is a certain kind of necessity, or at least a deductive consequence, from H to O.



Hypothesis and Evidence

An argument about the independence of an hypothesis from its evidence (or from evidence generally), or from observational consequences, can be articulated by saying that given exactly the same evidence, various and many hypotheses can explain that same evidence. Basically, this is a way of making the obvious point that evidence and hypothesis are not the same thing. Alternatively, an hypothesis is more than the evidence which supports it. (This is basically a rephrasing of the idea that “theory is always underdetermined by all available and relevant evidence”.)


In fact, just as I stated the truism that hypothesis and all available and relevant evidence are not the same thing, and also that rival hypotheses can explain the same evidence, so too it is the case that all these rival hypothesis - which are fighting to explain the same evidence - are not equal or identical either. And that lack of identity or equality, again, has nothing to do with the relevant evidence (which is the same for all the rival hypotheses).


It is commonly said, by both scientists and philosophers of science (though less by the latter), that these other factors include the degree of simplicity of the hypothesis as well as its explanatory power, esthetic value, comprehensiveness, etc. However, even though I have stressed the fact that evidence is not everything, it is, obviously, of vital importance. (How could it not be in science?) So it is true that many commentators, not always scientists, have stressed that Watson and Crick were esthetically delighted by the beauty of the double helix hypothesis for the structure of the DNA molecule. That's true. However, if they had wanted purely artistic pleasure they would have become painters or composers. That esthetic pleasure was largely generated by the simplicity of the double helix hypothesis. But simplicity, in scientific theory, is not an end in itself. That simplicity, in this and in many other cases, meant that there was a good chance that the said hypothesis is true/correct. That is, simplicity generated beauty and that beauty/simplicity generated the strong possibility of a correct hypothesis.

The Paradox of the Barber Who Shaves Everyone Who Doesn't Shave Himself

There are actually some (as it were) false paradoxes: arguments or situations which seem paradoxical until they are seen not to be paradoxical – simply false. Some logicians claim that this is one.



Well, does this barber shave himself? He must do because he shaves all those who don't shave themselves. But if he shaves himself, he can't be a member of [the class of those who don't shave themselves]. Yet his job is to shave all those who don't shave themselves – and he only becomes a member of that class when it's seen by himself that he doesn't shave himself.

Thus we have a contradiction. As a person who doesn't shave himself, he must shave himself. If he did shave himself, then he wouldn't be a member of [the Class of Those Who Don't Shave Themselves]. But the resultant situation, in both cases, is that he shaves himself. If he shaves himself, then obviously he shaves himself. However, if he doesn't shave himself, and he must shave all those who don't shave themselves, he must also end up shaving himself.

Why on earth should we assume that there isn't a barber who shaves all and only those who don't shave themselves? It's certainly not illogical if he excludes himself. But perhaps that's precisely the problem – he can't exclude himself! However, even if he can't exclude himself, the paradox doesn't appear to disappear: talk about not assuming that there could be a barber who only shaves those who can't shave themselves seems to be to sidestep it the problem.



The philosopher Roy A. I believe that Sorensen makes the mistake of saying “we should not assume that it is possible for there to be a barber who shaves all and only those he does not shave”. That locution doesn't appear to make sense. It's not paradoxical – just senseless.

The end result of this possible paradox is that we have a barber who shaves all and only those he does not shave. But I still can't work out how you get there. Therefore is may well be a mistake – perhaps just a typing mistake. The thing is, I don't know!

So perhaps Sorensen's locution is correct after all. The end result of this possible paradox is that we have a barber who shaves all and only those he doesn't shave. However, I still can't work out how Sorensen gets there. Therefore may well be a mistake – perhaps just a typing mistake.

David Hilbert

David Hilbert embarked on an enormous enterprise: the reduction of the whole of mathematics to a set of axiomatic systems. Clearly these systems must have been interrelated in many ways in order to determine and guarantee their mutual consistency. We can distinguish each axiomatic system by their different axioms. In addition, in order to move from these axioms to theorems, we must utilise something that is shared with logic: the rules of inference. Again like logic, one such rule of inference is the well-known modus ponens. This shared interest in the rules of inference is partly accounted for by the fact that "they are properties of reasoning as such". That is, both logic and mathematics must use the inferences which belong to every rational mind and even every thought or act of reasoning. They were, therefore, as foundational and fundamental as Aristotle’s ‘laws of thought’.

However, there must be some things, other than numbers, which distinguish mathematics from logic. For example, a prime candidate for this difference is the nature of the axioms in mathematics. They are different because they describe space, time and measurement in all its forms. These applications, to space, time and measurement clearly distinguish mathematics from logic, at least from pure formal logic but not, for example, quantificational logic.

Hilbert was a Platonist. He didn’t reject numbers in his systems. We cannot "eliminate the idea of number from the axioms". They are, then, fundamental to maths precisely because they are used in its many axioms. Hilbert expressed his Platonism in an even more platonistic way. Like Plato himself, "we must therefore suppose that numerical expressions stand for objects, which have a reality independent of our calculations". He was clearly not, therefore, a mathematical constructivist or a mathematical Wittgensteinian, and neither was he a mathematical Kantian. In addition, although these number-objects are "known to us through proof, but which are entities over and above the proofs by which we discover them". So Hilbert accepts that they are only known to us through proof, which is an operation and perhaps a psychological operation, these number-objects do not need us in order to exist. The numbers can then be called evidence-transcendent, at least in the case of those numbers and operations that can never be known, or are not known now. They are also mind-independent objects, even though they can be ‘known’ by minds.

In some cases, then, some object-numbers and operations may never be known as a matter of necessity. Take ‘Golbach’s theorem’ which cannot be proved (see Kripke’s 1971). These positions seem very counterintuitive to many minds, not only non-philosophical minds. Indeed they seem even stranger in the context of that branch of mathematical constructivism known as intuitionism. In the case of intuitionists, if a mathematical statement has not been proved or disproved it is, in fact, neither true nor false, mind-independently or otherwise. In fact, unproven statements have another ‘truth-value’: indeterminate. Incidentally, this is also the case with ‘future contingents’ – statements about the future. They too are neither false nor true, but they are indeterminate instead. Perhaps this is because a future-statement cannot be proved either, almost by definition.

In addition, being constructivists, intuitionists do not believe that numbers are objects either, whether abstract or concrete. We ‘construct’ numbers by the operations we carry out on them. They are not found, either, via platonic ‘intuition’ or Husserl’s ‘direct insight’. If there were no minds, there would be no numbers and no mathematical operations on these numbers.

'What do materialists make of love & justice?'

“How do you consider concepts (like love and justice) to be physical/material? Where do they exist? To say something like a number (which is abstract) is physical is misplaced concreteness.” - Wololo

I'm not sure that anyone who has sympathy with materialism, or physicalism, would say that love and justice were material in any strict sense. Nonetheless they may well give a physicalist or a naturalist account of such things. In depends on whether or not you have a Platonic or quasi-Platonic view on love and justice. If you do, then, by definition, they would be non-physical universals (or 'Ideas' in Plato's parlance).

However, if you aren't a Platonist of some kind, then love and justice can be given a naturalist, if not a physicalist, explanation which may or may not be successful. For example, love - rather than Love - can be explained in terms of human biology, history, human and social relationships, etc. – all of which a naturalistic in nature. It need not be the case that love and justice are 'reduced', strictly speaking, to such things. However, love and justice must be dependent on natural things and not run free, as it were, of them.

So, yes, both justice and love exist wherever there are examples or expressions of love or justice. Love can exist where acts of love occur and the same with justice. (For now we can forget the different positions people adopt on both love and justice because this question is about whether or not they are natural, or even physical, phenomena.)

No scientist today would ever say that numbers are physical or material. And only a very small numbers of philosophers would argue against numbers being abstract in nature. Despite that, these latter philosophers don't think that numbers are concrete either. Some philosophers argue that numbers, or equations, are basically invented. They are a product of conventions and symbols and the rules which are applied to those conventions and symbols. (Perhaps this does mean that numbers are concrete, in a certain sense.) In other words, there is no need for abstract numbers as such. These few philosophers, and even fewer mathematicians, argue that numbers are created, or invented, by the mathematical procedures that bring them about. That is, a new number is created when a new procedure brings that number into existence in a similar way in which a new concept or word (such as 'nerd') is brought into being. Or, if not words/concepts, numbers are created in the way that, say, a new design for a building is created. (I'm not saying I agree with any of this; only that these views exist.)