Frank Hutson Gregory

“ A logical reformulation The apples example can be reformulated as an experiment. Take a basket that contains a pair of apples and nothing else. Take a bucket that contains a pair of apples and nothing else. Empty the entire contents of the basket into the bucket taking care to make sure that everything that is in the basket goes into the bucket. Now how can we determine how many apples are in the bucket?
One way is to use the calculus of arithmetic. We can take the contents of the basket as an instantiation of the arithmetical notion "2". ”

“ There is another way to determine the number of apples in the bucket and this is by counting them. We can take an apple out of the bucket and say "one", then we can then take another apple out and say "two" and so forth. When there are no more apples left in the bucket we know we have counted them all. Let us call this the "counting method". The contention that arithmetic, understood in the constructivist sense, applies to reality is the contention that the calculation and counting methods will always give the same results. ”

“ It is, we might imagine, quite right that a man who has had four goats, but been unlucky and lost them all, should be given the same respect as a man who has only ever had three.
We need not assume that the goat-herds using either Tally System 1 or Tally System 2 have any knowledge of arithmetic. Nor need we assume that they can count in any way independently of their tally sticks. Beads are threaded and taken off as part of a public semi-religious ritual. Everybody in the tribe can agree when this ritual is properly performed. Children are taught the ritual along with various occult rituals. ”

“ Gaskin argues that an arithmetic formula such as "7 + 5 = 12" cannot mean the same as an empirical proposition such as would be obtained from counting groups of objects. He argues that in order to explain mistakes in counting we need to invoke the notion of counting correctly. But Gaskin argues the meaning of correct counting is dependent on logically true propositions of arithmetic. Therefore, empirical propositions based on counting do not have equivalent meaning, nor can they be used as equivalent substitutions for, arithmetical propositions. ”

“ Such a massive digression into philosophical logic can, for present purposes, be circumvented if the distinction between logical and factual truth is based on the key terms used to describe the difference between realism and contructivism. That is, logically true statements are those that are invented and what follows from them, factually true statements are those that are discovered to be true and what follows from them.
Following Popper we can say that all factually true universals are open to falsification. Therefore they are contingent. ”

“ Prima facie the propositions of applied mathematics appear to be about objects and events in the real world while those of pure mathematics do not.
If it were true that propositions of applied mathematics are about real world objects then this would suggest that they are justified empirically. Here we can identify two broad schools of thought. Following Tymoczko, accounts of the nature of arithmetic and mathematics can be described as realist or constructivist. "Realism assumes the reality of a mathematical universe which is independent of mathematicians who discover truths about this reality. ”

“ Each member of the tribe has a tally stick onto which beads are threaded. When a tribe member is given a goat, or when one of his goats gives birth, a new bead is threaded on to the owners tally stick. When one of his goats dies a bead is taken off the owner's tally stick. We can imagine that in the tribe social prestige and privilege is the determined by the number of goats that a person owns. Given this the tally system will be useful. It can be determined who has the most goats by placing different owners' tally sticks side by side. ”

“ All we need to say is that in any system for the calculation of real world quantities that employs logically true and non-falsifiable statements of arithmetic these statements can be replaced by a statement or statements that are not non-falsifiable statements of arithmetic. ”

“ Given that tally systems are contingent with regard to number, the logically true system of arithmetic is not only useful but essential. This might not be immediately apparent because it might seem that tally systems can be identified as systems of counting by the fact that the totals they produce stand in a one-to-one relation with objects in the real world. ”

“ If we are to determine that something is a member of a class there must be some criterion for class inclusion that we use to make the determination. Alternatively if we are to determine a criterion for class inclusion then that criterion must be true of all members of the class, therefore, in order to make the determination we must have identified the members of the class. It can be concluded that any useful term or class that has a logical extension must have a factual and contingent intension; and any term or class that has a logical intension must have a factual and contingent extension. ”

“ A realist account of pure mathematics and a constructivist account of applied mathematics.
All four permutations are open to immediate difficulties. The first needs to explain why arithmetic propositions do not appear to be falsifiable by experience. The second needs to explain why a mental construct, such as arithmetic, can be informative about reality while other analogous mental constructs, such as chess, are not. The third has similar difficulties, it needs to explain how a mental construct relates to a real world discovery. ”

“ Factually true premises can be introduced into an axiomatic system and theorems that follow from axioms and factual premises by means of the rules of production will inherit the contingency of the premises and be factually true. The problem is to determine why we need the logical truths. ”

“ Tally System 1 is meaningful quite independently of arithmetic formulas. However, as such correspondence does exist we are entitled to call it a system of counting. It is not that a logically true arithmetic is required in order to determine what correct counting is, as Gaskin suggested, but that a logically true arithmetic will enable us to determine what systems are to count as counting systems. Given arithmetic formulae as the logical intension of number it will be a matter of empirical inquiry and discovery which systems and rituals can be part of the extension i.e. ”

“ A more likely candidate is that he was saying that rules of inference are open to falsification by facts. I.e that if "All men are mortal" is a description that fits the world and "Socrates is a man" is a description that fits world, but "Socrates is mortal" is a description that does not fit the world, then it would be shown that modus ponens is not valid. ”

“ An animal is a snake if and only if it is a member of the viper family or cobra family or boa family or colubrid family or hydrophida family.
The players in a language game might assent to both statements without having decided which is a definition. Both are sufficient identification criteria. They therefore inhabit a logical limbo which will not be resolved until a particular fact forces the issue. ”

“ A fixed criteria for class membership will enable us to identify members of the class. If we define a tiger as a cat with stripes then we can say that if X is a cat and X has stripes then X is a Tiger. ”

“ E.g. every cat is a lion or a tiger or a leopard or a puma etc. On the basis of this we might formulate various factual hypotheses, i.e. that only cats have claws and that all cats have sharp teeth. These could lead to other factual universals e.g. that anything that has claws also has sharp teeth. Thus, extensive definitions can be instrumental in formulation of factual universals. ”

“ The definition of number The way in which number is defined can now be considered. One possibility is to define number in terms of arithmetic formulae. An intensive definition of the number "three" can be as follows: " (1 + 1 + 1) & (1 + 2) & (2 + 1) ". ”

“ Mathematics is a large subject and it not obvious that it is completely homogenous. The idea that parts of mathematics are invented and parts discovered should not discounted out of hand. However, if the inquiry is limited to basic propositions of arithmetic i.e. ”