It would be extremely bald on my behalf to disqualify certain interpretations of logic (such as math), given that within philosophy mathematics is one of the fields in which a genius can only stand aside and criticize certain theories with the out most perspective.
However, I will attempt none the less to look at several philosophical propositions that derive from a mathematical understanding and decide whether I agree or not.
Bertrand Russell was very keen on understanding arithmetic as a form of logic and Frege wasn't very convinced given that certain ideas of classification can consider arithmetic to be very abstract and not empirical whatsoever.
The example given was 'one drop of water + one drop of water = one drop of water (in bigger size)'. One drop of water (1q) + (1q) should equal to 2q since the principle of addition is clear but instead its 1Q (Q in capitals representing a lager substance in its same quantity).
Therefore the proposition of arithmetical abstract is clear.
My proposition is that this interpretation is erroneous, I will not refute that arithmetic can be considered abstract instead of empirical but this proposition in particular is incorrect and it demonstrate that Frege's ideas of sense vs reference are very much interconnected in all fields of theory of knowledge.
First of all we must look at the 'drop of water' a drop of water is a name given to a recognizable amount of water in a very small and somewhat inconclusive size of water with a cylindrical perimeter that evokes the idea of a small portion of water that has been removed or dis-attached itself from a larger much more dense amount of water. Once we recognize the drop with quantify it, in this case 1. The mistake lies in the reasoning behind this categorization of water with an arithmetic understanding such as (Quantity 1) when in itself this quantity cannot really be applied given its physical condition which is at a liquid state.
There is a reason why we measure water in liters and not in grams given its molecular structure and mass, therefore when we quantify its mass capacity we find ourselves using a different categorical group than a lemon or a house for example.
When it comes to a drop of water the identification is much less vague given that there could be a drop of water of one cubic centimeter and a drop of water of 0.5 millimeters, they are both the same thing and yet they aren't given that their size is completely different. Of course, in a mathematical approach their size is not important, 1 small house and 1 big house will still equal 2 houses regardless of the size, however we have no doubt of what a house consists of and its dimensions allow us to understand straight away how a house looks like whereas with water, (once again because of its composition) we cannot apply the same concept we can only assume it is water but we can't really identify it as a group such as 'a drop' of water because ultimately the drop means that 'it is a much smaller proportion of water that belongs to a larger group of water'
The arithmetic rule of water can only be applied when the substance is contained within a solid mass with sufficient space, because water with itself cannot be added, one drop of water in itself cannot be quantified as the name itself (Frege's rule of understanding a conceptual case scenario based on the names and classification group) proves that it is only a proportion of something much bigger, therefore once this smaller proportion is added to something of the same type it will increase in size rather add one more.
Water will react with itself in the same way that two solids will compose something bigger or a specific compound, the same way as I can say, add 2 legs, 2 arms, 1 torso, 1 head = 1 person.
My point is the following, the 'drop of water' theory (as I call it) that proves that arithmetic is an abstract form of logic rather than empirical is not sufficient and through Frege's understanding of language it can easily be refuted as a valid theory.
Moreover, for the sake of the argument we are not applying the concept that two solids cannot occupy the same space at the same time except a liquid in a subtle way, its molecular structure is far more divided and scattered and it allows molecules of water fill other gaps, increasing its size, not its quantity.
