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I've had at least two pure mathematics majors people argue strenuously with me that statistics, being applied mathematics, isn't really math, but that isn't really my point here.

That was extremely silly of them for a bunch of reasons, in addition to calling (applied) mathematics not mathematics:

• What is considered applied mathematics and what is considered pure mathematics is arbitrary cultural stuff and varies from community to community.
• Applied math is not fundamentally different from pure math, and relevant fields - regardless of whichever branches a particular culture considers to be 'applied' and not 'pure' - are studied the same way (in the relevant in this context sense) and with the same standards of rigour.

This is especially amusing to me - a person who has studied as both an applied mathematician and as a pure mathematician, - as some of my classmates from all of the relevant groups specialised in the field of probability theory and statistics.

Statistics is math that is derived from empirical observations

It isn't.
You can use statistics for empirical studies, but it itself is not studied empirically.

Without that actual data, statistics as pure mathematics is completely meaningless to the engineering process

Just as arithmetic, just as geometry, just as logic, just as stuff like control theory, wavelet theory, theories of differential and integral equations, vector field theory, theory of switch functions (not sure what the English name of that one is, actually), etc.
What you are probably confused about is that relevant fields can be used for models used by engineers, physicists, chemists, biologists, medical professionals, etc., and then jump to the conclusion that they must be studied empirically.

I'm saying there is a materialist dialectic that proceeds between the empirical, physical, observations made, and how math is then used to then depict, formulate and transform those observations into more empirical measurements, which then continues to transform those depictions

Okay? But how are engineers going to do their job without both the knowledge discovered by mathematicians and without other people's discoveries made using said knowledge? How would transistors come to be developed without relevant understanding of math, for example?

Quite often these days the math precedes the observation, but particularly before the advent of electronic computers it wasn't uncommon for the observation to precede the mathematics. For example, you absolutely cannot have transistors without the Fournier transformation, but you also cannot have transistors without the observed phenomenon and concept of electrical conductivity (or more usually talked about resistance), which was not originally conceptualized mathematically at all, though is now.

Firstly, that example does not actually show 'observation preceding the mathematics'.
Secondly, that example doesn't actually show how that development could be done without math. The fact that it also required empirical study and did not come about from pure reason is irrelevant if you are trying to claim that development and manufacture of devices that allowed the person I was initially responding to to make their comments could realistically be done without math.

Math is not dealing with literal objects

It is.
For example, again, mathematicians study such objects as numbers, functions, sets, propositions, transforms (and their invariants).

For example, 1+1=2 doesn't have to reference any kind of object at all in order to be self-contained, logical and true

Not sure what you are trying to say there.
The expression '1+1 = 2' refers to the proposition that some object referred to with the expression '1+1' (as we know, that is - in the standard context - a real number that is the successor of 1 in terms of Peano axioms) is the same object as the one referred to with the expression '2'. We could go into more detail here, but there are plenty of objects being referred to here.
What does 'self-contained' mean in this context? What does it have to do with references to any objects? What does it have to do with the rest of this conversation?
What does 'logical' mean? That the relevant expression refers to a proposition (which contradicts your claim that no objects are being referenced)? I genuinely do not know what the word is supposed to mean (other than 'not stupid' in a colloquial sense, which is not really applicable here).

This can be explicitly shown in things like statistics, where you have a mathematically logical statement that is 'people in the U.S. have on average 1.5 children'. A nonsense statement if taken purely empirically, but the idea of how an average is mathematically created can make it logically sound, however it is also completely meaningless to us if that average wasn't generated from real data.

I genuinely can't tell what you mean by 'taken purely empirically' without completely changing these two sentences to the point of the loss of any relevance to the conversation. Would you mind rephrasing that?
But also, not only is that not meaningless if it is not 'generated from real data', as that expression does have a meaning that I'm fairly confident is commonly understood, and that understanding of that meaning is not dependent on whether or not it is based on 'real data' (the data may be false, or it might not even exist to begin with) and whether or not one is presented with that data.
But also, not sure how this is supposed to be an argument against any of what I have said to begin with.

It is conceptualized mathematically now, but it didn't start as that, it didn't start as a formula, it came from some other ideas like 'lets make a more durable iron' and through our physical interaction with it has come to a point of conceptualized mathematics,...

Again, I am not sure how this is supposed to be an argument against me claiming that relevant stuff was discovered/developed using knowledge about the objects that are studied in mathematics-as-an-academic-field.
Do you think that I claimed that engineering or physics, or other relevant fields do not engage in empirical studies? If so, then I'm pretty sure that I can even quote myself saying the opposite in this thread.

Scientists and their benefactors wouldn't build the Hadron collider if just knowing the math was the answer

I never claimed that all that engineering and other relevant fields involve is just mathematics. What I did claim is that they all do use mathematics (not to the exclusion of empirical studies and knowledge developed through those).

but I am also arguing that mathematics likely isn't the end-all be-all understanding of accurate conceptualization

If by that you just mean that knowledge about objects that are studied in math-as-an-academic-field is not sufficient for stuff like engineering, physics, chemistry, etc., then I don't think you have been contradicted here by anybody - not by me, at least.