The title as many may know, is a quote from Wittgenstein. It is one that has haunted me for many years. As a first year undergrad, I had mistakenly enrolled in a second year course that was almost entirely based on Wittgenstein’s  Tractatus. Alarmingly, the drop date had passed before I grasped I was supposed to understand (at least some of) the Tractatus to pass. That forced me to repeatedly re-read it numerous times. I did pass the course.

However, I now think the statement is mistaken. At least, outside mathematics in subjects where what is being said is an attempt to say something about the world – that reality that is beyond our direct access. Here some vagueness has its place or may even necessary. What is being said will unlikely be exactly right. Some vagueness may be helpful here in the same way that sheet metal needs to stretch to be an adequate material for a not quite fully rigid structure.

Now, what I am thinking about trying to say more clearly at present is how diagrammatic reasoning, experiments performed on diagrams as a choice of mathematical analysis of probability models utilizing simulation, will enable more to grasp statistical reasoning better. OK, maybe the Wittgenstein quote was mostly click bait.

My current attempt at saying it with 4 paragraphs:

Grasping statistics more thoughtfully and more widely may be achieved by trying to understand experiments themselves more thoroughly. Experiments understood as making observations and making sense of them in terms of what to repeatedly expect in future observations. Now, as indicated above, experimental reasoning can be extended to include mathematics. That being the manipulation of diagrams or symbols taken beyond doubt to be true – experiments performed on abstract objects rather than chemicals – diagrammatical reasoning. An abstract diagram is made, manipulated and observed to understand the diagram as thoroughly as one can. It is simply a choice of method of mathematical analysis.
Now, probability models can be represented as diagrams and understood by experimenting on them by repeatedly simulating outputs from them. Given most statistical methods are based on probability models (though sometimes only implicitly) statistical methods are likely best grasped by that means. It is simply a choice of method of analysis.  So to more fully grasp statistical methods one needs to first fully grasp probability models. It’s all about sense making in statistics being formalized in assessments of what would repeatedly happen in some reality or possible world. That “what would repeatedly happen” is most explicitly defined in probability models.
Here, the proposed approach offers two advantages. The first being avoiding the need to learn formulas and their manipulations as well as rules for their use. The second being the ability to experiment with statistical methods in fake worlds where the underlying truth can be set and known. In this setting, the study of the performance of statistical methods and especially what to make of them might be easier to grasp. The intent is to enable some amount of do it yourself verification of statistical recommendations to avoid having to just take someone’s word for it.
No formulas will be required, just probability diagrams and repeated simulation. However, any real learning of something new is a painful struggle (there are likely biological reasons for that). One needs to develop an experimental attitude and persistent curiosity about how things really are. Devoting time to both defining more and more realistic like possible worlds. Using appropriate compositions of probability diagrams. Then gaining sufficient experience in these designed fake worlds where the truth is known. The challenge then becomes the application of that knowledge that in the real world where the truth is uncertain. That is transporting what is learned in the fake worlds to making sense of what is observed in this world.