What is extreme finitism?

 

Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects.

Strict Finitism is a form of finitism that rejects objects known as ‘countably infinite’.

Extreme Finitism goes much further as it rejects all uses of the concept of ‘infinity’.

Extreme Finitism is the philosophy that only accepts the existence of finite objects and where some symbols are allowed to represent process definitions where no end point has been specified. Furthermore, Extreme Finitism opposes the use of words that imply that ‘infinity’ is a valid concept.

These words include ‘infinitely many’, ‘an infinite number of’, ‘infinite’, ‘to infinity’, ‘continuous’, ‘analogue’, ‘forever’, ‘eternally’, ‘infinitesimal’ and all other words and phrases that imply ‘infinity’.
 

Extreme Finitism views a quantity as merely a property of something that does/could exist. A quantity is not an object in itself and it cannot be said to have its own existence. Suppose someone picks up two apples and discovers one has a rough texture and the other has a smooth texture. The textures have no inherent existence of their own; they are merely properties of other things that exist, namely the apples. We perform an examination/measurement process on the apples to determine the type of texture.

Similarly we could perform an examination of how many apples someone is holding, and our examination process might return a value of 2. Here we have a defined group that is ‘things the person is holding’, and our number 2 is a count returned from the examination/measurement process that ‘counts apples’ for the defined group. If the person does not exist, then we cannot perform the examination/measurement and so the corresponding numeric property cannot be evaluated. So, in the same way that texture is a property of something that exists, what we call a ‘natural number’ is a property of a defined group that exists.

Fundamental to modern mathematics is the claim that numbers have their own existence, and that mathematics is detached from physical reality. If we accept these claims, it can appear logical to say that if a natural number (n) exists then its successor (n+1) must also exist, thus the amount of natural numbers cannot be finite. We appear to have used logic to deduce that infinitely many of something is a valid concept. The basis of this dubious logic is that things like natural numbers can be said to ‘exist’ and that such objects are somehow detached from all physical reality.

A number appears to have a physical presence when we write one down (it physically exists on the paper), or when we store one on a computer (it physically exists in the state of voltages), or even when we think about one (it physically exists in the state of our brain chemistry). Therefore it could be argued that a number stored on a computer, say, has its own physical presence and as such has its own existence. A physical memory location might hold a value corresponding to a count of 2, but arguably this is just a description of a generic property. Similarly we could store the text “rough texture” on the computer, but this does not prove that a rough texture can have its own existence independent of all physical reality.

Perhaps our starting assumption should be that the only thing that exists is a shared physical reality that we refer to as the real world. Anything we imagine is just the result of brain chemistry, and just like computer memory, brain chemistry is a finite physical entity. Neither a biological brain nor an electronic computer can imagine (or visualise or picture) infinitely many of something, but perhaps they can delude themselves into accepting that they can? Perhaps we should not accept that we can imagine or think-of or otherwise conceive of anything that has no relation to the real world, perhaps we can only imagine that we can (in other words, perhaps the belief that we can conceive of non real world things is self delusion)? Perhaps we should re-invent mathematics from scratch, where every concept in our new mathematics has a basis in reality. Then we would never have to ask children to believe they can imagine what the square root of minus 1 is. Instead we might ask them to visualise a grid with the four directions ‘+’ (right), ‘-‘ (left), ‘+i’ (up), ‘-i’ (down) and a movement-based concept of multiplication so that they can easily visualise the movement sequences (corresponding to i squared and minus i squared) that reach the ‘one-square-in-the-left-direction’ position. This re-invention of mathematics should contain no paradoxes, no divide by zero issues, and no claims that we can work with actual infinities. In a new mathematics we could have a geometry based on a tangible definition of a point instead of a definition that says a point has no parts to it (currently a point is effectively defined as being nothing!). A natural number could be a measurable quantity instead of an abstract concept that has no connection with reality. Perhaps we should not even use the word ‘mathematics’ because maybe the fundamentals are really all about computing?

If we assume numbers have their own existence then we soon run into problems like the issue of ‘divide by zero’ and its affect on algebra. Then as mathematical concepts are supposedly not of this world, we don’t have any real-world examples that could help us to unravel the issue. And since everything is supposedly ‘abstract’ (a word misused to imply not of this world), then instead of saying “oops, divide-by-zero is a serious problem so we must have made a mistake in our reasoning somewhere”, what we actually concluded was “oh great, we have discovered a new rule. It says we are not allowed to divide by zero”. Then we dubiously claimed that everything was once again consistent now that we had included this newly discovered rule into ‘mathematics’.

For over two thousand years we have continued to interpret serious fundamental problems as discovered rules about what we can or cannot do. Fairly recently, Russell’s Paradox was handled by this approach. It was decided we can get around the paradox by following rules that prevent us from encountering the paradox. Perhap the most fundamentally of all, the problems arising from the claimed infinite divisibility of idealised lengths are simply called paradoxes and the accepted approach is to either ignore them or to admire them for being wonderful examples of how strange and mysterious mathematics can be! Then we claim everything is once again consistent and cannot be proven to be otherwise. Strangely only a small minority of people are concerned about this ‘make it up as you go along’ approach to mathematics fundamentals.

A common objection to finitism is to ask if there are not infinitely many natural numbers, then what is the last number? This argument relies on both sides agreeing that numbers have their own inherent existence. But if both parties agreed that a natural number refers to a generic quantity where a quantity is merely a property of a well defined group, and that a natural number has no existence of its own, then both parties would realise why it makes no sense to ask what is the last number.

Knowing how to count and knowing there is no defined end point in the (counting) algorithm does not make ‘infinity’ a reality. We can write down instructions on how to count natural numbers starting from one. Lots of different people can look at these instructions and start counting. The instructions might not specify when to stop counting and so in this sense there is no ‘last number’. However, only a finite number of people can ever follow these instructions and each can only count up to a finite number. Everything about this algorithm and its use is entirely finite. We might represent this algorithm as the series 1 + 1 + 1 +… or as the sequence 1, 2, 3, 4, … This demonstrates that having a counting algorithm with no end point specified does not imply that an actual infinity can/must somehow occur, despite what our intuition might tell us.

Click here to read about the background investigation.