The math here is just simple explicit examples of the general digital principle - Digital transform=nor-formula.: any consistent transform of digital input to digital outputiis equivalent to a finite temporal propositional logic formula(nor gates) common to computers.
That is, all consistent typed responses to typed input can be performed by a finite computer memory!
Arguments that cannot be put in a nor-formula are misleading. The only transform digital I/O that cannot be equaled by a nor-formula gives different(inconsistent) values at different times.

Name the input wires, I, and the output wires, O, of a nor-formula N.
I will write N(i)=o where i and o represent particular values - binary numbers - particular high or low voltages on the wires I and O.
In the computer, all inputs are distinct physical wires which may have humanly observable number names. [For simplicity, inputs and outputs will be considered on distinct wires - often for practicality, the same wires are used for input or output but 'enable' wires are used to keep them activated at separate times.]
For explicit temporal logic, N(i)=o will require the voltages represented by i to be stable at time T[i] and will require the voltages represented by o to be stable at time T[o]>T[i].  Also one should recognize that i and o are stable for only the requisite finite duration,  explicitly omitted here for brevity.

It is recognized that logic for modeling material objects, needs to materialize the same finite time and space properties. Supernatural or Platonic mediums involving timeless, infinite processes bring great risk of unmaterialistic implications. It is misleading conservative platonism, forced into logic history by the cave evolved unobservable human thought process, to say one has somehow discovered timeless, placeless truth. Any supernatural infinity that is fully measured, modeled and defined by a natural finity is thereby natural and finite. Those presumed aspects of infinity or higher logic that are programmable in a finite memory, independent machine are thus finite and low level. The popular Turing 'machine' with infinite memory or infinite speed is not a material existing machine. And human intuition based 'theorems' about it do not rigorously apply to real computers. They are certainly not super-rigorous, since they are not made in a completely observable, independent logic machine. My human definition of a finite process is the number or complete presence of nor-gate activations in a given completely observable, independent mechanism, before the process starts repeating results. By 'independent machine' I mean a human may give input to a humanly constructed machine but it is then left alone to construct its transform of that input. The human may observe the process with an oscilloscope, logic analyzer or monitor of high enough impedance so as not alter the results.
If that number of activated nor-gates is important, special hardware could automatically get this number for the observer or the computer. See interactive Recursive finite automatons have finite cycles 10k. However, this could cause a problem: a finite memory computer can not totally 'look at itself'. the special hardware feeds a new number back into the computer for the current number of activated nor-gates which then causes more activations and thus an addition to that total. So this hardware must deactivate itself once it responds to a request and thus remain incomplete.

Sets:

• Input wires are particular wires that are connected to a logic circuit N.

N(i)=o represents the output voltage pattern o resulting on the output wires O of the nor formula N whose input wires I have had the input voltage pattern i for a time long enough for the physical circuit in N to settle to its defined stable state.
• Concatination(a+b) refers to all voltages on all the wires represented by A and B which can be given a single name C for all these wires.
• Set s of N is all those different binary numbers m for which N(m)=s.
• Nor-formula Equality: N(a)=N(b), not equal N(a) != N(b).

function unique(f)// remove duplicates
{var s="";var z="";var j=1;var a="a";var b="b";var c=f+" ";var x=f.length//
for(j=0; j for(var i=j+1; i if(a==b){c=c.substring(0,i)+c.substring(i+1,x);x--;i--}//
}//repetitions?
}//firsts
return c;//
}//unique
function Union(fA,fB){var c="";var c=fA+fB; var d=""//concatinate inputs into one
d=unique(c); //
return d}//union
function Intersection(fA,fB){var s="";var z="";var j=1;var a="";var b="";var c="";var maxa=fA.length; maxb=fB.length//
var c=fA;var FA=unique(c); c=fB; var FB=unique(c);c="" for(j=0; j if(b==" "||b=="")break//
}//over b
}//over a
return c}//Intersection
function runc(f){var c="";fA=f.fa.value; fB=f.fb.value;f.fI.value= Intersection(fA,fB); f.fU.value= Union(fA,fB); //Union(f) // runD(f) }

Material Relations:

• reflexive N,e:N(a+a)=e. This is to be interpreted or programmed: given nor-formula N, the output is e for any double input a.
• transitive N: If N(a+b)=e and N(b+c)=e then N(a+c)=e.
• symmetric N: If N(a+b)=e then N(b+a)=e.
• asymmetric N,e: If N(a+b)=e then N(b+a) != e.
• antisymmetric N,M:  If N(a+b)=e and N(b+a)=e then M(a)=M(b).
• connected N,M,e: If M(a) != M(b) then N(a+b)=e or N(b+a)=e
• right unique N,M:  If N(a+c)=e and N(b+c)=e then M(a)=M(b).
• left unique N,M:  If N(a+c)=e and N(a+b)=e then M(c)=M(b).

Material Math Induction:

Definition:
In a given finite machine, 2 digital transforms(nor-formulas) p(n,pni,pnr), q(n,qni,qnr) are called equal for a range of binary values, n1<=n<=n2, where 'in' is the nth initialization time (the time when the nth input n is stable) and rn is the nth result time, (rn>in when output of nor-formula is stable) provided:

• Simultaneously: n at same time as n+1
1. p(n1,pin1,rn) = q(n1,qin1,rn) for initialization times so the results of the transforms are the same at time rn.
2. if p(n,pni,rn) = q(n,qin,rn) then p(n+1,pin+1,rn+1) = q(n+1, qin+1,rn+1) independent of n.
• Sequentially: n+1 chronologically after n
1. p(n1,pin1,rn1) = q(n1,qin1,rn1) for initialization times so the results of the transforms are the same at time rn1.
2. if p(n,pin,rn) = q(n,qin,rn) then p(n+1,rn,rn+d) = q(n+1, rn,rn+d) independent of n, d>0. Note this recognizes the n+1 case may be physically caused later at a time, T[rn+d], by the nth case and this natural fact may be used in the proof.

A variant of the above is obtained by replacing '<' by '>' and '+' by '-'. So induction runs backwards.
Another variant is a simpler 2. 2'. p(n,pni,rn) = q(n,qin,rn) independent of n.
Another variant is to index the nor-formulas instead of the arguments, which would be fixed.

Finally, the above may be generalized beyond necessarily using ordered binary numbers. The 2nd condition's 'n+1' may just indicate a new, previously unused input, not the next.

Example application, a+b=b+a:
Naturally, I detail a materially implemented proof for known finite electronic adders, I'm not trying to make a traditional Platonic proof designed for unknown human intelligence. [incomplete. Javascript planned]

In the following, c:carry bit, a, b:bits of added binary numbers, 'xor(a,b)=1':only if {a=1,b=0} or {a=0,b=1}, 'and(a,b)=1': only if {a=1,b=1}, 'or(a,b)=1': only if {a=1,b=1} or {a=0,b=1} or {a=1,b=0}.
For induction proof, we consider circuit module for nth and next stage(without time symbols): xor(cn,(xor(an,bn))=sumn, or(and(an,bn),and(cn,xor(cn,xor(an,bn)) )=c(n+1).
With time symbols: xor(cn,(xor(an,bn,xin,xrn),xrn,x2rn)=sumn, or(and(an,bn,oin,orn),and(cn,xor(cn,xor(an,bn,xin,orn),orn,aorn),aoin,x2rn )=c(n+1).
Examples where OLD math induction does not work but NEW does:

1. 'stack' memory:
   A 'stack' memory for storing only 2 values is considered. That is, if x is stored, then y, then z, only y and z remain, x being discarded. For OLD traditional math induction that ignores time, it's clear the stack can store zero, and if it stores n it can store n+1[by pushing n-1 out]. But the traditional implication that the stack therefore stores[for all time] all the integers is obviously false. I must say that I skimmed over the 'n+1' as it requires the space for the 2 stored values of the stack to be unrealistically unbounded, but the paradox is based on the number of values.
   For materialistic math induction time is accepted as important and instrumental. It thus makes no paradox in the stack  It merely verifies any two values selected from a given finite memory computer may be stored at one time in the stack. It would do this by verifying the circuit logic for copying values from a given sized memory into the stack.
2. 'unbounded' memory:
   A real operating computer has a physical volume that is increased as memory modules are added. One may humanly imagine being able to always add more memory as needed, but the speed of electrons or photons being finite starts putting greater and greater delays on getting the data from one end to the other. Also the whole gets so massive that it becomes a black hole.