I started writing this with intention of posting a short informative article about aluminum in deodorant. Several weeks and one deep hole of research later, here is the result. It is, I think, quite informative but alas, not short. If you'd like to skip the close examination of research, feel free to scroll right down to "Fear and Marketing." You can also take a look at the source list at the bottom, where I've noted a few articles which are fairly quick, easy reads and are from reputable sources. The rest of you, lets see if we can sort through the fluff and get down to brass tacks!
First of all, lets make sure we're on the same page as far as terminology. The aluminum we're talking about is not elemental aluminum but rather aluminum salts, which is aluminum combined with another element or elements. This is because aluminum salts are the only ingredients approved by the FDA for use as an antiperspirant. Why is this important? Well, the properties of the different elements can change radically, depending on what they are bonded to. Pure sodium is explosive when mixed with water, pure chlorine is a toxic gas. Combine them in the right way and you get sodium chloride, table salt!
The other term to define is deodorant. Aluminum salts are specifically used as an antiperspirant, something that stops you from sweating. Many deodorants are also antiperspirants but some are not. My product, Farmer Pits, is only a deodorant and thus does not contain any antiperspirants (or aluminum). In fact, since aluminum salts don't fit most people's definition of "natural" pretty much every natural deodorant is aluminum free and are often advertised as such.
Aluminum is an abundant element on earth and occurs naturally in both food and water. Everyone consumes some amount of aluminum every day but the body does not absorb much of it through digestion (about 0.3% from water and 0.1% from food ). Also, the body can eliminate aluminum through the urine, though at a fairly slow rate. Aluminum toxicity is definitely a risk with high exposure but this is something that typically happens to those with occupational exposure to aluminum welding fumes or aluminum processing dust 
Links to Alzheimer's?
High aluminum exposure has also been linked to dementia symptoms, which led some to make a connection between aluminum and Alzheimer's disease. This began when some long term kidney dialysis patients began developing dementia symptoms. After much testing, doctors discovered that the cause was too much aluminum in the "dialysate" (one of the fluids used to perform dialysis). Because the aluminum was going directly into the bloodstream, the body was absorbing it at a much higher rate than it would through food and began to cause these dementia symptoms.  People exposed to high levels of aluminum through their job exhibit similar neurological symptoms.
(As an aside, one of the only reasons I could find that doctors would recommend avoiding aluminum in deodorant was in people with low kidney function or kidney failure. Healthy kidneys can safely excrete the amount of aluminum absorbed from deodorant but unhealthy ones can't.)
However, these symptoms of aluminum exposure, while similar to Alzheimer's on the surface, are not the same disease and further testing has been unable to prove a link between aluminum exposure and Alzheimer's risk. Some studies have shown "very uncertain indications of an association between aluminum exposure and Alzheimer’s disease"  So unfortunately, as is sometimes the case with science, we're just not entirely sure, although the evidence does seem to point towards safety in this case. Personally, I would want to see a lot stronger evidence of danger from aluminum before I started to worry about it.
Links to breast cancer?
Aluminum compounds have been used in deodorant products since the early 1900s.  The absorption rate of aluminum through the skin is even lower than through food, ranging from 0.01% up to 0.06% when the skin was damaged . So it's safe to say that antiperspirants are not exposing you to high enough amounts of aluminum to cause the kind of acute toxicity we've talked about so far. But another connection you've probably heard about is that between aluminum in breast cancer.
Two observations that encouraged scientists to pursue research into aluminum and breast cancer were first, that more tumors were found in the tissue under the armpit near where deodorant is applied and that higher levels of aluminum were found in tissue samples from women diagnosed with breast cancer.  This encouraged a lot of studies into the connection but what we now think is that since the tissue near the underarm is a denser, glandular tissue, tumors are more likely to develop there anyway. Tumor tissue also tends to store aluminum (as well as other minerals) at a higher rate than other tissues, which explains the higher concentrations in tumor samples. 
As with many scientific studies, news coverage about the dangers of aluminum was incomplete. Over the years there have been hypotheses about its danger to our health and a few studies that have shown weak connections to Alzheimer's and breast cancer. After the big scary headlines subsided and further study showed no evidence of danger, aluminum wasn't in the news anymore. Now, the National Cancer Institute lists the connection between breast cancer and antiperspirants under its "Common Cancer Myths and Misconceptions" category and says, "no scientific evidence links the use of these products [antiperspirants and deodorant] to the development of breast cancer." 
Fear and Marketing
Ok, so here you are on my hippie, crunchy natural soap and deodorant website reading about how aluminum in deodorant is pretty ok actually...what's going on? Well, for starters, I'm not about to start putting aluminum in my deodorant! I think Farmer Pits already works great for most people and natural deodorants are about more than just avoiding aluminum. There are all kinds of things like parabens and antibacterials in the bigger brand deodorants that I haven't covered here and frankly I don't have the time to do more research! Plus, my recipe is easy and low tech enough that I can make it in my house without factory sized machinery, so its a great fit for a small home business.
Something that has always irked me about advertising by companies presenting themselves as "natural" and "green" is the use of fear. They talk about the big brand products having scary chemicals and ingredients that you can't pronounce, they talk about how much cleaner, healthier and purer you can be if you use their products instead. For me, that kind of advertising is an immediate turn off. Secret and Old Spice may be full of chemicals that I would want to research further before using them regularly but at least their advertising boils down to "it smells nice and it works."
When you see marketing claims for a product, don't be afraid to do your own research. I think we are starting to get jaded in this day and age about what an amazing tool the internet is for research. Certainly there is a lot of fluff and straight up garbage to sort through, so I like to focus foremost on articles put out by universities that cite their sources. From there you can usually read the articles they cite yourself and find links to further sources from there. Wikipedia can even be a good place to start, though you may have to do a little more sifting to find quality sources.
Doing research really takes the fear factor out of the scary headlines, I find. There are so many things that we are constantly told "cause cancer" or "promote inflammation" and any number of other buzz words. But do just a little background reading and you'll often find that these claims are overblown or straight up baseless. It's also good to remember that science is an evolving process. Theories are tested and changed as new experiments provide new evidence. Was the theory that aluminum might cause Alzheimer's or breast cancer a stupid one? Of course not. It fit the evidence that was available at the time. But if we're going to think scientifically, we have to be willing to be wrong in light of new evidence, rather than cling to old theories with misplaced loyalty.
Perhaps my unwillingness to pull on the heart string of health and purity is why I've never been that great at marketing. It seems to almost be a requirement for natural products. Instead I will continue on the path of providing information that is as well-researched and unbiased as possible. Because if you sweat buckets and work in an office right next to other people, it could be that Farmer Pits isn't for you, in which case I hope you can use antiperspirant without fear or guilt and will perhaps try some of my soaps or salves instead!
So don't use my products out of fear that your life will be nasty, brutish, short and cancer-ridden  if you dare apply a chemical antiperspirant to your armpits. Use my products because they smell nice and they work.
Image of a block of aluminum: By Unknown author - http://images-of-elements.com/aluminium.php, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=9084427
https://www.accessdata.fda.gov/drugsatfda_docs/omuf/OTC%20Monograph_M019-Antiperspirant%20Drug%20Products%20for%20OTC%20Human%20Use%2011.23.2021.pdf (just a dry list of FDA approved antiperspirants)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5651828/ (I used this one A LOT, a bit dense but very informative)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC494487/?page=1 (even denser but great detail on the issue of aluminum toxicity in kidney dialysis)
https://www.herbalix.com/assets/EFSA_Study_-_European_Food_Safety_Authority.pdf (a nice short look at aluminum exposure through food and water)
https://www.cancer.gov/about-cancer/causes-prevention/risk/myths/antiperspirants-fact-sheet (The National Cancer Institute, short, easy read)
https://www.pennmedicine.org/updates/blogs/health-and-wellness/2019/june/deodorant (a much less dense read than all the medical studies)
https://wexnermedical.osu.edu/blog/aluminum-deoderant (another more digestible and shareable article)
https://www.gutenberg.org/files/3207/3207-h/3207-h.htm (if you ever wondered where that "nasty, brutish and short" quote comes from it's from here. Not recommended as light, feel good reading)
https://sercblog.si.edu/science-superstition-and-the-goose-barnacle/ (I really wanted to work in an analogy about the medieval belief that a certain species of goose came from barnacles, or possibly trees, rather than eggs but I couldn't work it in)