# References for the notion of grounding, applied to mathematical truths

I am interested in papers that discuss the notion of grounding and applies it to mathematical statements. For example, the facts that 1+1=2 and 2+2=4 collectively ground their conjunction 1+1=2 AND 2+2=4. Also, for instance, the facts that 0+0=0, 0+1=1, 0+2=2, etc collectively ground the universal statement "For all x, 0+x=x". Basically, I am interested in texts that define a notion of grounding for mathematical truths. It can't simply be that a true mathematical statement A grounds statement B if and only if A materially implies B, for all true statements of mathematics imply each other. Also, I am not necessarily requiring that the grounding relation is irreflexive. In my mind, it is perfectly legitimate for some statements to be their own grounds. For example, in my view at least, the axioms of a mathematical theory ground themselves. Anyway, are there such books or papers or texts that talk about grounding but restricted to mathematical truths, perhaps even defining the grounding relation?

• Off the top of my head, this is what the standard theory of well-founded sets (i.e. ZFC) amounts to. IIRC the SEP article on grounding mentions philosophers who read the metaphysical grounding relation off the mathematical well-founding relation, but I see no reason why you could not read things in the other direction. Commented Feb 10, 2022 at 0:43
• Quickly googling finds papers like: "Ground and Explanation in Mathematics" by Lange, "On Grounding Arithmetic" by Ciro De Florio, "Grounding in Mathematical Structuralism" by John Wigglesworth. Commented Feb 10, 2022 at 14:35
• In Robinson's Q "∀x.x+0=x" is simply grounded by nothing but one of its axioms directly, however, surprisingly re your 0+0=0, 0+1=1...collectively ground ..."∀x.0+x=x", the conclusion is undecidable in Q per Godel's incompleteness. Thus even to ground such obvious and ideal mathematical "fact" is hard and illusory as Shurangma sutra explained: With your own mind, you grasp at your own mind. What is not illusory turns into illusion. If you don’t grasp, there is no non-illusion... Commented Feb 10, 2022 at 22:57
• Lange, Ground and Explanation in Mathematics:"there is currently no widely accepted account of either mathematical explanation or grounding... I will try to stick with features of grounding that are relatively uncontroversial among grounding theorists". Also, Poggiolesi-Genco, Conceptual (and hence mathematical) explanation, conceptual grounding and proof Commented Feb 11, 2022 at 1:30

I'm not aware of such precise thing, but this might provide a starting guide.

To start, let's say that the final ground of all reason is logic. Logic is not a set of disparate statements, but an interrelated structure of truth (surely tautological), grounding being the very structure, also tautological (there's no rule that confirms that the truth of logic is a final truth). Interrelated implies the necessary existence of systemic relationships between concepts, that's why `1=1 ^ 2=2`, the conjunction being an explicit indication that both concepts `1=1` and `2=2` are in the same structure of truth.

But perhaps the answer that fits best is the approach of the discipline of formal systems. Formal systems, formally (using a precise language and concepts) determine a scope of such structure, which is to be considered the ground of a specific discipline of knowledge. So, such domain determine the concepts and axioms that must be followed to any further analysis. Upon such structure, new elements can be produced, by means of theorems and conclusions.

There's a key point to consider: in order to grasp the axioms and concepts of a formal system (learn them, know them, analyze them, expand them), at least human experience is mandatory (yes! this means using shoes, walking to the school, scratching the nose, laughing...). In most cases, much more than that: knowing some specific branch of mathematics, chemistry, etc. It would be literally impossible to provide the necessary knowledge of what is a dot to a martian without having lived on earth... for years.

Consider this: any computer can get the dictionary relations in seconds, because it is just a tautological set of self-references, it is a circular set of definitions. But in order to understand a simple word, knowledge of all feelings, knowledge of things, the environment, mental states, culture, other human beings, etc. are mandatory.

So, below the ground of any formal system, there is an implicit load of human experience that is required. Notice that here the problem becomes metaphysical, an approach that few want to consider.

Noticing that all mathematics, logic and science are completely linked to human subjectivities is bad news for most (that's probably the most relevant problem in current philosophy, and perhaps it starts to be so in science, since QM), so, that's why the topic is almost never developed. I personally say that metaphysics is the elephant in the room of science. The problem is there, in front of our eyes, it is huge, nobody is searching for solutions; so, better not to mention it.

• Re. "let's say that the final ground of all reason is logic" - They could be viewed as all aspects of the same, i.e. briefly pointing to: "ratio speaks as "grounds" as well as "Reason." and : "Just as the bifurcated word ratio passes over into the basic words of modern thinking—"Reason" and "grounds"—so a Greek word speaks in the Roman word ratio: it is called λόγος." - from Heidegger's The Principle of Reason, page 105. So I'd say the grounding is the logic and the reasoning, (and some je ne sais quoi). Commented Mar 1 at 22:45

I am interested in papers that discuss the notion of grounding and applies it to mathematical statements.

I have found few scholarly articles by just searching for "grounding mathematics": of course, those contain further references.

For example, the facts that 1+1=2 and 2+2=4 collectively ground their conjunction 1+1=2 AND 2+2=4

As for any "grounding", I think the notion at play is that a statement is true if we can provide evidence for it: which is the approach of "constructive logic" but, more generally, it is the Curry-Howard correspondence as it applies not only to types and theorems, but also to computations and proofs.

For example, the evidence needed to prove `A /\ B` is, by definition of conjunction, exactly a pair of evidence for `A` and evidence for `B`.

Also, for instance, the facts that 0+0=0, 0+1=1, 0+2=2, etc collectively ground the universal statement "For all x, 0+x=x".

No, that "collectively" is just not defined apriori, especially considering that we cannot provide an infinite amount of evidence: i.e., more precisely, proofs must be finite, by definition! So, here (when domains become infinite) is where, more than just plain evidence, we need an induction principle.

The bottom line, it seems to me (hoping I am not simplifying too much), is that all mathematical ground boils down to (mathematically) inductive ground!